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/■ary Sytttm 
ity of Wisconsin-Maditoil 
otate Street 
dison, Wl 53706-1484 
S.A. 




A MANUAL 



OF 



RULES, TABLES, AND DAT4 

FOR 

MECHANICA^ lENGINEERS 

BASED ON THE MOST RECENT INVESTIGATIONS: 

OF CONSTANT USE 
IN CALCULATIONS AND ESTIMATES RELATING TO 

Stkskcth of Matbkials AMD OP Elementary Constructions; Labour; 
Heat and its Applications, Steam and its Properties, Combustion and Fuels, 
Steam Boilers, Steam Engines, Hot-Air Engines, Gas-Engines ; Flow op Air and op 
Water; Air Machines; Hydraulic Machines; Mill-gearing; Friction and the Resistance of 
m achinbry, &c ; weights, measures, and monies, british and forbign, with the reciprocal 
Equivalents for the Conversion of British and French Compound Units op 

Weight, Pressure, Time, Space, and Moi^BV : Specific Gravity and ^ 

THE Weight op Bodies; Weight op Metals, &c. 

with 
Tables of Logarithms, Circles, Squares, Cubes, Square-Roots, and Cube-Roots; 

AND MANY OTHER USEFUL MATHEMATICAL TaBLBS. 

BY 

DANIEL KINNEAR CLARK, 



% I 



AUTHOR OF 



MEMBER OF THE INSTITUTION OP CIVIL ENGINEERS; 
RAILWAY MACHINERY," "EXHIBITED MACHINERY OF 1862," BTC. 



SECOND EDITION. 




* LONDOVsT: 

BLACKIE & SON: PATERNOSTER BUILDINGS, 

GLASGOW AND EDINBURGH. 

1878. 
A/i Rights Reset ved. 



QhABQOWl 

W. O. BLAOKIS AJSD CO., PBINTEBS, 

VILLAFIXLD. 









PREFACE. 



This Work is designed as a book of general reference for Engineers : 

— ^to give within a moderate compass the leading rules and data, 

with numerous tables, of constant use in calculations and estimates 

relating to Practical Mechanics. The Author has endeavoured to 

concentrate the results of the latest investigations of others as well 

as his own, and to present the best information, with perspicuity, 

conciseness, and scientific accuracy. 

Amongst the new and original features of this Work, the follow- 
ing may be named : — 

In the section on Weights and Measures, the weight, volume, 
and relations of water and air as standards of measure, are concisely 
set forth. The various English measures, abstract and technical, 
are given in full detail, with tables of various wire-gauges in use: 
and equivalent values of compound units of weight, power, and 
measure — as, for example, miles per hour and feet per second. 
The French Metric Standards are defined, according to the latest 
determinations, with tables of metric weights and measures, equi- 
valents of British and* French weights and measures, and a 
number of convenient approximate equivalents. There is, in addi- 
tion, a full table of equivalents of French and English compound 
units of weight, pressure, time, space, and money — as, for example, 
pounds per yard and kilogrammes per metre; which will be found 
of great utility for the reciprocal conversion of English and French 
units. 

The tables of the Weight of bars, tubes, pipes, cylinders, plates, 
sheets, wires, &c., of iron and other metals, have been calculated 
expressly for this Work, and they contain several new features 
designed to add to their usefulness. They are accompanied by a 
summary of the various units of weight of wrought iron, cast iron, 
and steel, with plain rules for the weight. 

In the section on Heat and its Applications, the received mechan- 
ical theory is defined and illustrated by examples. The relations 
of the pressure, volume, and temperature of air and other gases. 



vi PRfeFACE. 

with their specific heat, are investigated in detail. The transmission 
of heat through plates and pipes, between water and water, steam 
and air, &c., for purposes of heating or cooling, is verified by many 
experimental data, which are reduced to units of performance. 

The physical properties of steam are deduced from the results 
of Regnault's experiments, with the aid of the mechanical theory 
of heat A very full table of Ihe Properties of Saturated Steam is 
given. The table is, for the most part, reproduced from the article 
** Steam," contributed by the Author to the Encyclopedia Britannicay 
8th edition, and it was the first published table of the same extent, 
in the English language, based on Regnault's data. An original 
table of the properties of saturated mixtures of air and aqueous 
vapour is added. 

In the section on Combustion, new and simple formulas and data 
are given for the quantity of air consumed in combustion, and of 
the gaseous products of combustion, the heat evolved by combus- 
tion, the heating power of combustibles, and the temperature of 
combustion ; with several tables. 

On Coal as a Fuel, both English and Foreign, its composition, 
with the results of many series of experiments on its combustion, 
are collected and arranged. Thequantityof air consumed in its com- 
bustion, and of the gaseous products, with the total heat generated, 
are calculated in detail. Coke, lignite, asphalte, wood, charcoal, 
peat, and peat-charcoal, are similarly treated ; whilst the combus- 
tible properties of tan, straw, liquid-fuels, and coal-gas, are shortly 
treated. 

The section on Strength of Materials is wholly new. The great 
accumulation of experimental data has been explored, and the most 
important results have been abstracted and tabulated. The results 
of the experiments of Mr. David Kirkaldy occupy the greater por- 
tion of the space, since he has contributed more, probably, than any 
other experimentalist to our knowledge of the Strength of Materials. 
The Author has investigated afresh the theory of the transverse 
strength and deflection of solid beams, and has deduced a new and 
simple series of formulas from these investigations, the truth of which 
has been* established with remarkable force by the evidence of experi- 
ment These investigations, based on the action of diagonal stress, 
throw light upon the element called by Mr. W. H. Barlow, "the 
resistance of flexure:" revealing, in a simple manner, the nature 
of that hitherto occult entity; and showing that flexure is not the 
cause, but the effect of the resistance. In addition to formulas 



PREFACE. vil 

for beams of the ordinary form, special formulas have been 
deduced for the transverse strength and deflection of railway 
rails, double-headed or flanged, of iron or steel; in the estab- 
lishment of which he has availed himself of the important 
experimental data published by Mr. R. Price Williams, and by 
Mr. B. Baker. To our knowledge of the strength of timber, 
Mr. Thomas Laslett has recently made important additions, and the 
results of his experiments have been somewhat fully abstracted and 
analyzed. But woods, by their extremely variable nature, are not 
amenable, like wrought-iron and steel, to the unconditional applica- 
tion of formulas for transverse strength. The Author has, never- 
theless, deduced from the evidence, certain formulas for the trans- 
verse strength and deflection of woods, with tables of constants, 
which, if applied with intelligence and a knowledge of the uncer- 
tainties, cannot fail to prove of utihty. 

The Torsional Strength of Solid Bodies has also been investigated 
afresh, and reduced to new formulas. 

In dealing with the Strength of Elementary Constructions, the 
Author has brought together many important experimental results. 
In treating of rivet-joints and their employment in steam-boilers, 
he has, he believes, clearly developed the elements of their strength 
and their weakness. By a close comparison of the results of tests of 
cast-iron flanged beams, it is plainly shown that the ultimate 
strength of a cast-iron beam is scarcely affected by the proportionate 
size of the upper flange, and that the lower flange' and the web are, 
practically, the only elements which regulate the strength. The 
tests of solid-rolled and rivetted wrought-iron joists are also ana- 
lyzed ; and for the strength and deflection of these, as for those of 
cast-iron flanged beams, new and simple rules and formulas are 
given, A new investigation, with appropriate formulas, is given 
for the bursting strength of hollow cylinders, of whatever thickness. 
It is shown that the variation of stress throughout the thickness, 
follows a diminishing hyperbolic ratio from the inner surface to- 
wards the outer surface. The resistance of tubes and cylindrical 
flues to collapsing pressure is also investigated, and formulas based 
on the results of experience are given. 

On the subject of Mill-gearing, a new and compact table of the 
pitch, number of teeth, and diameter of toothed wheels is given, 
with new formulas and tables for the strength and horse-power of 
the teeth of wheels, and for the weight of toothed wheels. New 
formulas and tables are given for the driving power of leather 



viii • PREFACE. 

belts, and the weight of cast-iron pulleys. For the strength of 
Shafting, — cast-iron, wrought-iron, and steel, — a new and complete 
series of formulas has been constructed, comprising its resistance 
to transverse deflection and to torsion, with very full tables of the 
weight, strength, power, and span of shafting. 

The Evaporative Performance of Steam-boilers is exhaustively 
investigated with respect to the proportions of fuel, water, grate- 
area, artd heating-surface, and the relations of these quantities are . 
reduced to simple formulas for different types of boilers. The 
actual evaporative performances of boilers are abstracted in tabular 
form, comprising those of the Lancashire and the Galloway boilers 
at Wigan, tested by Mr. Lavington E. Fletcher. 

The Performance of Steam worked expansively, in single and 
in compound cylinders, is exhaustively analyzed by the aid of 
diagrams ; the similarity and the dissimilarity of its action in the 
Woolf-engine and the Receiver-engine, are investigated; and the 
principles of calculation to be applied respectively to these, the 
leading classes of compound engines, are explained. The best 
working ratios of expansion are deduced from the results of numerous 
experiments and observations on the performance of steam-engines. 

The principles of Air-compressing Machines, and Compressed-air 
Engines, — a branch of mechanical practice of comparatively recent 
origin, — are investigated, and convenient formulas and tables for 
use are deduced. 

Such are some of the new features of this volume. It may be 

added, that the other portions of the Work, likewise, have been 
carefully elaborated. The whole of the materials for its preparation 
have been drawn from the best available sources, foreign as well as 
English. Vast stores of the results of experience are accumulated 
in the Proceedings of t/te Institution of Civil Engifieers^ the 
Proceedings of tfie Institution of Mechanical Engineers, as well as 
in various periodicals and journals English and foreign. From 
these and other acknowledged sources, the Author has drawn 
much of his material. 



D. K. CLARK. 



8 Buckingham Street, Addphi, 

London, 20th March, 1877. 



CONTENTS. 



GEOMETRICAL PROBLEMS. . 

PACB 

Straight Lines — Straight Lines and Circles— Circles and Rectilineal Figures — The 
Ellipse — The rarabola — The Hyperbola — The Cycloid and Epicycloid — 
The Catenary — Circles — Plane Trigonometry — Mensuration of Surfaces — 
Solids — Heights and Distances, I 

MATHEMATICAL TABLES. 

Explanation of the following Tables : — 32 

Logarithms of Numbers from i to 10,000, 38 

Hyperbolic Logarithms of Numbers from 1. 01 to 30, 60 

Numbers or Diameters of Circles, Circumferences, Areas, Squares, Cubes, 

Square Roots, and Cube Roots, .66 

Circles : — Diameter, Circumference, Area, and Side of Equal Square, . 87 

Lengths of Circular Arcs from 1° to iSo**, 95» 97 

Areas of Circular Segments, 100 

Sines, Cosines, Tangents, Cotangents, Secants, and Cosecants of Angles, . . 103 

Logarithmic Sines, Cosines, Tangents, and Cotangents of Angles, . . 1 10 

Rhumbs, or Points of the Compass, 117 

Reciprocals of Numbers from I to 1000, - 118 

WEIGHTS AND MEASURES. 

Water as a Standard — Weight and Volume of pure Water — The Gallon and other 
Measures of Water — Pressure of Water — Sea-water — Ice and Snow — French 
and English Measures of Water, 1 24 

Air as a Standard — Pressure of the Atmosphere — Measures of Atmospheric Pres- 
sure — ^Weight of Air — Volume — Specifie Heat, 127 

Great Britain and Ireland — Imperial Weights and Measures, .128 

Measures of Length : — Lineal — Land — Nautical— Cloth, 129 

Wire-gauges, 130 

Inches and their Equivalent Decimal Values in part^of a Foot — Fractional Parts 

of an Inch, and their Decimal Equivalents, 135 

Measures of Surface : — Superficial — Builders' Measurement— Land, . .136 

Measures of Volume : — Solid or Cubic — Builders* Measurement, . • *37 

Table of Decimal Parts of a Square Foot in Square Inches, . . .138 

Measures of Capacity : — Li<|uid— Dry — Definition of the Standard Bushel — Coal 

— Old Wme and Spirit — Old Ale and Beer — Apothecaries* Fluid, . 138 

Measures of Weight: — Avoirdupois — Troy — Diamond — Apothecaries* — Old 
Apothecaries' — Weights of Current Coins — Coal— Wool — Hay and Straw 
— Com and Flour, 140 

Miscellaneous Tables : — Drawing Papers — Commercial Numbers — Stationery — 
Measures relating to Building — Commercial Measures — Measures for 
Ships, 143 

Comparison of English Compound Units : — ^Measures of Velocity — Of Volume 

and Time — Of Pressure and Weight — Of Weight and Volume — Of Power, 144 



X CONTENTS. 

PAGE 

France — The Metric Standards of Weights and Measures — Metre— Kilogramme, . 146 
Countries where the system is legalized, . . . . . • « • .146 

Measures of Length, 147 

Wire-gauges, 148 

Measures of Surface, 149 

Measures of Volume : — -Cubic — Wood, 149 

Measures of Capacity : — Liquid — Dry, 149 

Measures of Weight, 150 

Equivalents of British Imperial and French Metric Weights and Measures, . 150 

Measures of Length — Tables of Equivalent Values of Millimetres and Inches — 
Square Measures or Measures of Surface — Cubic Measures — Wood Mea- 
sure — Measures of Capacity — Measures of Weight, 150 

Approximate Equivalents of English and French Measures, 156 

Equivalents of French and English Compound Units of Measurement : — Weight, 
Pressure, and Measure — Volume, Area, and Length — Work — Heat — Speed — 

Money, 157 

German Empire : Weights and Measures : — Length — Surface — Capacity — Weight, 160 

Values of the German Fuss or Foot in the various States, 161 

Old Weights and Measures in Prussia (Kingdom of) — Bavaria (Kingdom of) — 
Wiirtembeig (Kingdom of) — Saxony (Kingdom of) — Baden (Grand-duchy of) 
— Hanse ']u)wns: — Hamburg — Bremen — Lubec — Old German Customs 

Union — Oldenburg — Hanover, &c., 162 

Austrian Empire, 170 

Russia, 171 

Holland — Belgium — Norway and Denmark — Sweden 173 

Switzerland — Spain — Portugal — Italy 175 

Turkey — Greece and Ionian Islands — Malta, 178 

Egypt — Morocco — Tunis — Arabia — Cape of Good Hope, 179 

Indian Empire — Bengal — Madras — Bombay — Ceylon, 180 

Burmah — China— Cochin-China — Persia — ^Japan — ^Java, 183 

United States of North America, 186 

British North America, 187 

Mexico — Central America and West Indies — West Indies (British) — Cuba — Guate- 
mala and Honduras — British Honduras — Costa Rica — St. Domingo, . . 187 

South America — Colombia — Venezuela — Ecuador — Guiana — Brazil — Peru — Chili — 

Bolivia — Argentine Confederation — Uruguay — Paraguay, .... 188 

Australasia : — New South Wales — Queensland — Victoria — New Zealand, &c., . 189 

MONEY— BRITISH AND FOREIGN. 

Great Britain and Ireland : — Value, Material, and Weight of Coins — Mint Price of 

Standard Gold, &c., 190 

France: — Material and Weight of French Coins, and Value in English Money, . 190 

German Empire : — Names and Equivalent Values of Coins, 191 

North and South Germany (Old Currency of), 191 

Hanse Towns (Old Monetary System of): — Hamburg, Bremen, Lubec, . . . 191 

Austria — Russia — Holland — Belgium — Denmark — Sweden — Norway, . . . 192 

Switzerland — Spain — Portugal — Italy — Turkey — Greece and Ionian Islands — Malta, 1 93 

Egypt — Morocco — Tunis — Arabia — Cape of Good Hope, 194 

Indian Empire — China — Cochin-China — Persia — ^Japan — Java, .... 195 

United States of North America, 195 

Canada — British North America, 196 

Mexico— West Indies (British) — Cuba — Guatemala — Honduras — Costa Rica — 

St. Domingo, 196 



CONTENTS. XI 

PACK 

Sooth America — C olombia — Venezuela — Ecuador — Guiana — BrazU — Peru — Ch ili 

— Bolivia — ATgentine Confederation — Uniguay — Paraguay, . . . igt 

Austialasia, . . - 197 

WEIGHT AND SPECIFIC GRAVITY. 

Standard Bodies and Temperatures for Comparative Weight — Rules for Specific 

Gravity, u 198 

General Comparison of the Weights of Bodies, •199 

Tables of the Volume, Weight, and Specific Gravity of Metallic Alloys — Metals — 

Stones, 200 

Coal — Peat — Woods — Wood-Charcoal,* 206 

Animal Substances — Vegetable Substances, • . 212 

Weight and Volume of various Substances, by Tredgold, 213 

Weight and Volume of Goods carried over the Bombay, Baroda, and Central 

Indian Railway, 213 

Weight and Specific Gravity of Liquids, . 215 

Weight and Specific Gravity of Gases and Vapours, 216 

WEIGHT OF IRON AND OTHER METALS. 

Data for Wrought Iron — for Steel— for Cast Iron, . .' . . . .217 

Tables of Weights: — Weights of given Volumes of Metals — Volumes of given Weights 
of Metals — Weight of One Square Foot of Metals — Weight of Metals of a 
given Sectional Area, 218 

Special Tables for the Weight of Wrought Iron: — 

Rales for the Weight of Wrought Iron — Cast Iron — and Steel, , . . 223 

Rule for the Length of I cwt. of Wire of different Metals, of a given thickness, 224 

Weight of French Galvanized Iron Wire, 225 

Special Tables of the Weight of Wrought-Iron Bars, Plates, &c. ; Multipliers 
for other Metals : — Flat Bar Iron — Square Iron — Round Iron — Angle Iron 
and Tee Iron — Wrought-Iron Plates — Sheet Iron — Black and Galvanized- 
Iron Sheets — Hoop Iron — Warrington Iron Wire — Wrought-Iron Tubes, 
by Internal Diameter — Wrought-Iron Tubes, by External Diameter, . 226 

Weight of Cast Iron, Steel, Copper, Brass, Tin, Lead, and Zinc — Special Tables : — 
Cast-Iron Cylinders, by Internal Diameter — Cast- Iron Cylinders, by External 
Diameter — Volumes and Weight of Cast-Iron Balls, for given Diameters; 
Multipliers for other Metals — Diameter of Cast- Iron Balls for given Weights, 253 

Weight of Flat-Bar Steel — Square and Round Steel — Chisel Steel, . . . 259 

Weight of One Square Foot of .Sheet Copper — Copper Pipes and Cylinders, by 
Internal Diameter — Brass Tubes, by External Diameter — One Square Foot 
of Sheet Brass, 261 

Siie and Weight of Tin Plates — Weight of Tin Pipes and Lead Pipes — Dimen- 
sions and Weight of Sheet Zinc, 268 

FUNDAMENTAL MECHANICAL PRINCIPLES. 

Forces in Equiubrium:— Solid Bodies — Fluid Bodies, 271 

Motion : — Uniform Motion — Velocity — Accelerated and Retarded Motion, . . 277 

Gravity; — Relations of Height, Velocity, and Time of Fall — Rules and Tables, . 277 

Accelerated and Retarded Motion in General: — General Rules— Descent 

on Inclined Planes, 282 

Mass, 287 

Mechanical Centres :— Centre of Gravity — Centre of Gyration— Radius of 
Gyration — Moment of Inertia — Centre of Oscillation — The Pendulum — 
Length of Seconds Pendulum — Centre of Percussion 287 

Central Forces : — Centripetal Force — Centrifugal Force, 294 



Xll CONTENTS. 

PAGB 

Mechanical Elements r—The Lever— The Pulley— The Wheel and Axle— 
The Inclined Plane — Identity of the Inclined Plane and the Lever — The 
Wedge — The Screw, 296 

Work:— English and French Units of Work — Work done by the Mechanical Ele- 
ments — By Gravity — Work accuxmilated in Moving Bodies — Work done by 
Percussive Force, 312 

HEAT. 

Thermometers: — Table of Equivalent Degrees by Centigrade and Fahrenheit — 

Pyrometers, 317, 967 

Movements of Heat: — Radiat;ion — Conduction — Convection, .... 329 

The Mechanical Theory of Heat: — Mechanical Equivalent of Heat— Joule's 

Equivalent in English and French Units — Illustrations, .... 332 

Expansion by Heat: — Linear and Cubical Expansion, 335 

• Table of Linear Expansion of Solids, 336 

Expansion of Liquids,^ . 338 

Expansion of Gases — The Absolute Zero-point — Table of the Compression of 

Gases by Pressure under a Constant Temperature, 342 

Relations of the Pressure, Volume, and Temperature of Air and other Gases — 
General Rules — Special Rules for One round weight of a Gas, with Table 
of Coefficients — Table of the Volume, Density, and Pressure of Air at 
various Temperatures, . * . . . 346 

Specific Heat: — Specific Heat of Water, with Table— Specific Heat of Air- 
Specific Heat of Solids — Specific Heat of Liquids — Specific Heat of Gases, . 352 

Fusibility or Melting Points of Solids: — Table, 363 

Latent Heat of Fusion of Solid Bodies, with Rule and Table, .... 367 

Boiling Points of Liquids, 368 

Latent Heat and Total Heat of Evaporation of Liquids, 370 

Boiling Points of Saturated Vapours under various Pressures, . • 37^ 

Latent Heat and Total Heat of Evaporation of Liquids under One Atmosphere, 372 

Liquefaction and Solidification of Gases, 372 

Sources of Cold: — Siebe's Ice- making Machine — Carre's Cooling Apparatus — 

Frigorific Mixtures, 373 

STEAM. 

Physical Properties of Steam, 378 

Gaseous Steam — Its Expansion — Its Total Heat, 383 

Specific Heat of Steam — Specific Density of Steam — Density of Gaseous Steam, 384 

Properties of Saturated Steam from 32' to 212* F., 386 

Properties of Saturated Steam for Pressures of from I lb. to 400 lbs., . 387 

Comparative Density and Volume of Air and Saturated Steam, . • 39^^ 

MIXTURE OF GASES AND VAPOURS. 

Respective Pressures of Gas and Vapours in Mixture, 392 

Hygrometry, 392 

Properties of Saturated Mixtures of Air and Aqueous Vapour, with Table, . 394 

COMBUSTION. 

Combustible Elements of Fuel — Process of Combustion, 398 

Air Consumed in the Combustion of Fuels :— Quantity of the Gaseous Pro- 
ducts of the Complete Combustion of One Pound of Fuel — Surplus Air, . 400 

Heat Evolved by the Combustion of Fuel : — Heat of Combustion of Simple 

and Compound Bodies — Heating Powers of Combustibles, .... 402 

Temperature of Combustion, 407 



CONTENTS. xm 



FUELS. 

PAGE 

Pods or Combustibles generally used, . « 409 

Coal: — Its Varieties — Small Coal: — Its Utilization — Washing Small Coal — 

Deterioration of Coal by Exposure to Atmosphere, 409 

British Coals — Composition of Bituminous Coals — Dr. Richardson's Analysis, . 412 

Weight and Composition of British and Foreign Coals, by Delab^che and 

Playfair, 413 

Weight and Bulk of British Coals, 416 

Hygroscopic Water in British Coals, 416 

Torbanehill or Boghead Coal, with Table of its Composition, . . . > 417 

American and Foreign Coals : — Composition, Weight and Bulk, . . .418 

French Coals: — Utilization of the Small Coal — Composition of French Coals — 

Mean Density, Composition, and Heating Power, 420 

Indian Coals : — Australian and Indian Coals — Composition, ..... 423 

Combustion of Coal : — Process of Combustion — Gaseous Products of the Com- 
bustion of Coal — Surplus Air — Total Heat of Combustion of British Coals, . 426 

CoK£ : — Proportion of Coke from Coals — Anthracitic Coke — Weight and Bulk of 

Coke — Composition of Coke — Moisture in Coke — Heating Power of Coke, . 430 

LiCNiT£ AND Asphalts : — Density, Composition, and Heating Power of Lignites 

and Asphaltes, 436 

Wood: — Moisture in Wood — Composition — Weight and Bulk of Wood, with 
Table — Firewood— Quantity of Air Chemically Consumed in the Complete 
Combustion of Wood — Gaseous Products — Total Heat of Combustion — 
Temperature of Combustion, . . 439 

Wood-Charcoal : — ^Yield of Charcoal — Composition, with Table of Composition 
at various Temperatures — Carbonization of Wood in Stacks, and Yield of 
Charcoal — Manufacture of Brown Charcoal — Distillation of Wood — Oiiarbon 
de Paris (artificial fuel) — Weight and Bulk of Wood-Charcoal — Absolute 
Density of Charcoal — Moisture in Charcoal — Air Consumed in the Combus- 
tion of Charcoal — Gaseous Products — Heat of Combustion, . . . 444 

Peat: — Nature and Composition — Condensed Peat — Average Composition — Pro- 
ducts of Distillation — Heating Power of Irish Peat, 452 

Peat-Charcoal: — Composition and Heating Power, 455 

Tan : — Composition and Heating Power, , 455 

Straw: — Composition, 456 

Liquid Fuels : — Petroleum, Petroleum-Oils, Schist Oil, and Pine-wood Oil ; their 

Composition and Heating Power, 456 

Coal-Gas : — Composition and Heating Power, . . . ... • • 457 

APPLICATIONS OF HEAT. 

Transmission of Heat through Solid Bodies — from Water to Water 
THROUGH Solid Plates and Beds: — M. Peclet's Experiments— Mr. James 
R. Napier's Experiments — Circumstances which affect the Ratio of Trans- 
misaon — Mr. Craddock's Experiments, 459 

Heating and Evaporation of Liquids by Steam through Metallic 
Surfaces: — Experiments by Mr. John Graham, by M. Clement, by M. 
Pcclet, by MM. Laurens and Thomas, by M. Havrez, by Mr. William 
Anderson, by Mr. F. J. Bramwell — Table of Performance of Coiled Pipes 
and Boilers in Heating and Evaporating Water by Steam, with Deductions, 461 

CoouNG of Hot Water in Pipes: — Observations of M. Darcy — Experiments 

by Tredgold — Deductions, 469 

Cooling of Hot Wort on Metal Plates in Air: — Results of Experiments 

at Trueman*s Brewery, 470 

CoouNG OF Hot Wort by Cold Water in Metallic Refrigerators: — 

Table of Results of Performance, and Deductions, 471 



XIV coNTE^r^s. 

PAGE 

Condensation of. Steam in Pipes Exposed to AiR:—Expcriments by Tred- 
gold, and by M. Burnat, on Pipes with various Coverings, with Table — 
Experiments by Mr. B. G. Nichol, by M. Clement, by M. Grouvelle — 
Condensation of Steam in a Boiler Exposed in Open Air, .... 472 

Condensation of Vapours in Pipes or Tubes by Water:— M. Audenet's 
Experiments on Steam — Mr. B. G. Nichol's Experiments — Condensation of 
other Vapours, 475 

Warming AND Ventilation: — Allowance of Air for Ventilation, . . . 477 

Ventilation of Mines by Heated Columns of Air.— Furnace- Ventilation 

— Mr. Mackworth's Data 479 

Cooling Action of Window-Glass:— Mr. Hood's Data, .... 480 

Heating Rooms by Hot Water: — Mr. Hood's Estimates — Total Quantity of 
Air to be Warmed per Minute — Table of the Length of 4-inch Pipe required 
to Warm any Building — Boiler-power — French Practice — Perkins' System, . 481 

Heating Rooms by Steam: — Length of 4-inch Pipe required — French Practice, 486 

Heating by Ordinary Open Fires and Chimneys:— M. Claudel's Data, . 488 

Heating by Hot Air and Stoves: — Sylvester's Cockle-Stove — French Prac- 
tice — House- Stoves placed in the Rooms to be Warmed — House- Stoves 
placed outside the Rooms to be W^armed, 488 

Heating of Water by Steam in Direct Contact: — Mr. D. K. Clark's 

Experiments, 490 

Evaporation (Spontaneous) in Open Air: — Dalton's Experiments, and Detluc- 

tions — Rule for Spontaneous Evaporation — Dr. Pole's Formula, . . '491 

Desiccation by Dry Warm Air: — Design of a Diying Chamber— Results of 
Experiments — Drying-house for Calico — Drying Linen and Various Stuffs — 
Drying Stuffs by Contact with Heated Metallic Surfaces — Drying Grain — 
Drying Wood, 493 

Heating of Solids: — Cupola Furnace — Plaster Ovens — Metallurgical Furnaces 

— IMast Furnaces, 497 



STRENGTH OF MATERIALS. 

Definitions 500 

Work of Resistance of Material, 501 

Coefficient of Elasticity, 503 

Transverse Strength of Homogeneous Beams, 503 

Symmetrical Solid Beams: — Investigation and Generalized Formula, . . 503 

Formula for the Transverse Strength of Solid Beams of Symmetrical Section, 
without Overhang, and Flanged or Hollow — For Unsymmetrical Flanged 

Beams — Neutral Axis — Elastic Strength, 509 

Forms of Beams of Uniform Strength:— Semi-Beams Loaded at One End 

I — Uniformly Loaded, 517 

Forms of Beams of Uniform Strength, Supported at Both Ends — Under a Con- 
centrated Rolling Load, 521 

Shearing Stress in Beams and Plate-Girders 525 

Deflection of Beams and Girders :— Investigation — Rectangular Beams— 

Double-flanged — Uniform Beams Supported at Three or more Points, . . 527 

Torsional Strength of Shafts:— Round— Hollow— Square— Deflection, . 534 

Strength of Timber:— Results of Experiments, 537 

Transverse Strength of Timber of Large Scantling, 542 

Elastic Strength and Deflection of Timber: — Experiments by MM. Chevandier 

and Wertheim, by Mr. Laslett, by Mr. Kirkaldy, by Mr. Barlow, . . 545 

Rules for the Strength and Deflection of Timber, 548 

Strength of Cast Iron: — Tensile Strength and Compressive Strength- Results 

of Experiments, 553 

Shearing Strength, 561 



CONTENTS. XV 

PAGB 

Tmnsvcrse Strength: — Results of Experiments — Test Bars — Transverse Deflection 

and Elastic Strength, . . / 561 

Torsional Strength, 565 

Strekgth of Wrought Iron :~Tensile Strength, &c.— Mr. Kirkaldy's Experi- 
ments, 567 

Experiments of the Steel Committee of Civil Engineers, 579 

Hammered Iron Bars (Swedish) — Knipp and Yorkshire Plates— Pi-ussian Plates, 581 

Iron Wire, 586 

Shearing and Punching Strength, 5S7 

Transverse Strength — Deflection and Elastic Strength, 588 

Torsional Strength, 590 

Strength of Steel:— Mr. Kirkaldy*s Early Experiments— Hematite Steel— 

Krupp Steel, 593 

Experiments of the Steel Committee, 596 

Experiments at H.M. Gun Factory, Woolwich — Fagersta Steel, Mr. Kirkaldy's 

Experiments, in seven series, 604 

Siemens- Steel Plated and Tyres — Mr. Kirkaldy's Experiments, . . .612 

Whitworth's Fluid- compressed Steel, 614 

Sir Joseph Whitworth's Mode of Expressing the Value of Steel, . .615 

ChemofTs Experiments on Steel, 616 

Steel Wire, 617 

Shearing Strength of Steel, 617 

Transverse Strength and Deflection, 617 

Torsional Strength, 619 

Strength Relatively to the Proportion of Constituent Carbon, . . . .621 

Resistance of Steel and Iron to Explosive Force > . 622 

Recapitulation of Data on the Direct Strength of Iron and Steel: — 
Tensile and Compressive Strength of Cast Iron, Wrought Iron, and Steel — 
Diagram of the Relative Elongation of Bars of Cast Iron, Wrought Iron, and 

Steel, 623 

Working Strength of Materials— Factors of Safety: — Factors of Safety 
for Cast Iron, Wrought Iron, Steel, and Timber — Load on Foundations, 

Mason-work — Ropes — Dead Load — Live Load, 625 

Tensile Strength of Copper and other Metals:— Tables of the Strength 

of Copper and its Alloys: Tin, Lead, Zinc, Solder, 626 

Tensile Strength of Wire of Various Metals:— Tenacity of Metallic 

Wires at Various Temperatures — Wires of Various Metals, . . 628 

Strength op Stone, Bricks, &c. : — ^Table of the Tensile Strength of Sandstones 
and Grits, Marbles, Glass, Mortar, Plaster of Paris, Portland Cement, Roman 
Cement, Granites, Whinstone, Limestone, Slates, Bricks, Brickwork in 

Cement — Adhesion of Bricks, 629 

STRENGTH OF ELEMENTARY CONSTRUCTIONS. 

Rivet-Joints: — In Iron Plates, 633 

In Steel Plates, 642 

PiLUiRS or Columns : — Compressive Strength 643 

Cast-Iron Flanged Beams: — Transverse Strength, 647 

Deflection and Elastic Strength, 652 

Wrought-Iron Flanged Beams or Joists: — Solid Wrought-iron Joists — 

Transverse Strength and Deflection, 653 

Riretted Wrought-iron Joists, 657 

Bt'CKLED Iron Plates, 660 



I 



XVI CONTENTS. 

PACB 

Railway Rails: — Transverse Strength of Rails of SynMnetrical Section, . .661 

Rails of Unsymmetrical Section, . 665 

Deflection of Rails, 668 

Steel Springs: — ^Laminated and Helical, 671 

Ropes: — Hemp and Wire, 673 

Chains, 677 

Leather Belting, 679 

Bolts and Nuts 680 

Screwed Stay-Bolts and Flat Surfaces 685 

Hollow Cylinders — Tubes, Pipes, Boilers, &c.: — Resistance to Internal or 

Bursting Pressure — Transverse Resistance, 687 

Longitudinal Resistance to Bursting Pressure, 692 

Wrought-iron Tubes, 693 

Cast-iron Pipe 693 

Resistance to External or Collapsing Pressure — Solid- drawn Tubes — Large Flue 

Tubes — Lead Pipes, 694 

Framed Work — Cranes, Girders, Roofs, &c.: — The Triangle the Funda- 
mental Feature, 697 

Warren-Girder Loaded at the Middle, and at an Intermediate Point — Uniformly 

Loaded — Rolling Load, 699 

Parallel 1-attice-Girder. 708 

Parallel Strut-Girder, 708 

Roofs, 713 

WORK, OR LABOUR. 

Units of Work or Labour: — Horse-power — Mechanical Equivalent of Heat 

— Labour of Men, 718 

Labour of Horses — Work of Animals Carrying Loads, 720 

FRICTION OF SOLID BODIES. 

Laws of Friction:-— Friction of Journals — Friction of Flat Surfaces, . . 722 

Friction on Rails: — M. Poiree's Experiments 724 

Work and Horse-power Absorbed by Friction:— Formulas, . . . 725 

MILL-GEARING. 

Toothed Gear:— Pitch of the Teeth of Wheels— Spur Fly-wheels— Toothed 

Wheels for Millwork — Rules, 727 

Form of the Teeth of Wheels, . . .731 

Proportions of the Teeth of Wheels, 734 

Transverse Strength of the Teeth of Wheels — Working Strength, . . . 735 

Breadth of the Teeth of Wheels, 737 

Horse-power Transmitted by Toothed Wheels, 737 

Weight of Toothed Wheels, . . 739 

Frictional Wheel-Gearing, 741 

Belt- Pulleys and Belts.-— Tensile Strength, 742 

Horse-power Transmitted by Belts, 743 

Adhesion and Power of Belts — Examples of very wide Belts, .... 744 

India-rubber Belting, 750 

Weight of Belt- Pulleys, 750 

Rope Gearing: — Transmission of Power by Ropes to Great Distances, . . 753 

Cotton Ropes^ . , , . , 755 



CONTENTS. XVll 

PAGB 

Shafting: — ^Transverse Deflection of Shafts, ,756 

Ultimate Torsional Strength of Round Shafts, 758 

Torsional Deflection of Round Shafts, 759 

Power Transmitted by Shafting 760 

Weight of Shafting, 761 

Strength and Horse-power of Round Wrought-iron Shafting, .... 762 

Frictional Resistance of Shafting, 763 

Ordinary Data for the Resistance of Shafting, ... ... 763 

Joomals of Shafts, 766 

EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. 

Normal Standards, 768 

Heating Power of Fuels:— Table of Heating Power, 769 

E%'APORATivE Performance of Stationary and Marine Steam - Boilers, 

WITH Coal: — Surplus Air Admitted to the Furnace, .... 770 

Experiments on the Evaporative Power of British Coals, by Delab^che and 

Playfair, 770 

Evaporative Performance of Lancashire Stationary Boilers at Wigan — With 
Economizer and Without Economizer ~ - Water-tubes — Temperature of 
the Products of Combustion, and of the Feed-water — Trials of D. K. 
Clark's Steam-Induction Apparatus — Of Vicars' Self- feeding Fire-grate, . 771 

Evaporative Performance of South Lancashire and Cheshire Coals in a Marine 

Boiler, at Wigan 781 

Trials of Newcastle and Welsh Coals in the Wigan Marine Boiler, . . 784 

Evaporative Performance of Newcastle Coals in a Marine Boiler, at Newcastle- 

on-Tyne, 785 

Trials of Newcastle and Welsh Coals in the Marine Boiler at Newcastle, for the 

Board of Admiralty, 787 

Trials of Welsh and Newcastle Coals in a Marine Boiler at Keyham Factory, . 790 

Evaporative Performance of American Coals in a Stationary Boiler, . . . 791 

Evaporative Performance of an Experimental Marine Boiler, Navy Yard, New 

York, 795 

Evaporative Performance of Stationary Boilers in France, .... 796 

Evaporative Performance of Locomotive Boilers, 798 

Evaporative Performance of Portable- Engine Boilers, 801 

Relations of Grate-Area and Heating Surface to Evaporative Per- 
formance: — Mr. Graham's Experiments — Experiments by Messrs. Woods 
and Dewrance — Experimental Deductions of M. Paul Havrez, . . . 802 

Formulas for the Relations of Grate-Area, Heating Surface, Water, 

and Fuel:— General Equations, 804 

Formulas for the Experimental Boilers, 807 

General Formulas for Practical Use, 819 

Table of the Equivalent Weights of Best Coal and Inferior Fuels, . . . 820 

STEAM-ENGINE. 

Action of Steam in a Single Cylinder:— The Work of Steam by Expan- 
sion — Clearance — Formulas for the Work of Steam — Initial Pressure in the 
Cylinder — Average Total Pressure in the Cylinder — Average Effective Pres- 
snre — Period of Admission and the Actual Ratio of Expansion — Relative 
Performance of Equal Weights of Steam Worked Expansively — Proportional 
Work Done by Admission and by Expansion — Influence of Clearance in 
Redndng the Performance of Steam 822 

Table of Ratios of Expansion of Steam, with Relative Periods of Admission, 

Pressures, and Total Performance, 835 

b 



XVlll CONTENTS. 

PACK 

Total Work Done by One Pound of Steam Expanded in a Cylinder, . 838 

Consumption of Steam Worked Expansively per Horse-power of Net Work 

per Hour, 840 

Table of the Work Done by One Pound of Steam of lOO-lbs. Pressure per 

Square Inch, 841 

Net Cylinder-Capacity Relative, to the Steam Expanded and Work Done in 

One Stroke, 843 

Table of ditto, 844 

Compound Steam-Engine; — Woolf Engine— Receiver-Engine— Ideal Diagrams, 
without Clearance — Work of Steam as Affected by Intermediate Expansion 
— Intermediate Expansion — Work, with Clearance — Comparative Work of 
Steam in the Wool! Engine and the Receiver- Engine, 849 

Formulas and Rules for Calculating the Expansion and the Work of Steam, . 869 
Compression of Steam in the Cylinder, 878 

Practice of the Expansive Working of Steam: — Actual Performance — 

Data — Deductions — Conclusions, 879 

FLOW OF AIR AND OTHER GASES. 

Discharge of Air through Orifices — Anemometer, 891 

Outflow of Steam through an Orifice, 893 

Flow of Air through Pipes and Other Conduits 894 

Resistance of Air to the Motion of Flat-Surfaces 897 

Ascension of Air by Difference of Temperature, 897 

WORK OF DRY AIR OR OTHER GAS, COMPRESSED OR 

EXPANDED. 

Work at Constant Temperatures:— Isothermal Compression or Expansion, 899 

Work in a Non-conducting Cylinder, Adiabatically, .... 901 

Efficiency of Compressed-Air Engines, 909 

Compression and Expansion of Moist Air, 912 

AIR MACHINERY. 

Machinery for Compressing Air and for Working by Compressed 
Air: — Compression of Air by Water at Mont Cenis Tunnel Works — By 
Direct-action Steam-pumps — Compressed-air Machinery at Powell Duffryn 
Collieries, 915 

Hot- Air Engines: — Laubereau's— Rider's — Belou*s — ^Wenham's, . . . 917 

Gas-Engines: — Lenoir^s — Hugon*s — Otto & Langen's, 920 

Fans or Ventilators: — Common Centrifugal Fan— Mine- Ventilators — GuibaFs 

Fan — Cook's Ventilator, 924 

Blowing Engines 926 

Root's Rotary Pressure-Blowers, 92^ 

FLOW OF WATER. 

Flow of Water through Orifices: — Formulas — Mr. Bateman's Experi- 
ments, 929 

Mr. Brownlee's Experiments with a Submerged Nozzle, 931 

Flow of Water Ovi;R Waste-Boards, Weirs, &c., 932 

Flow of Water in Channels, Pipes, and Rivers, 932 

Cast-Iron Water Pipes, 934 

C AST-Iron Gas Pipes, 936 



CONTENTS. • XIX 



WATER-WHEELS. 

PAGE 

Wheels on a Horizontal Axis:— Undershot- Wheels— Paddle- Wheels— Breast- 

Whcels — Overshot- Wheels, 937 

Wheels on a Vertical Axis:— Tub— Whitelaw*s Water-mill— Turbines- 
Tangential Wheels 939 

MACHINES FOR RAISING WATER. 

Pumps: — Reciprocating Pumps — Centrifugal Pumps — Chain Pump — Noria, . 944,. 968 

Water- works Pumping Engines, . . . . 948 

Hydzanlic Rams, 949 

HYDRAULIC MOTORS. 

Hydraulic Press, 950 

Armstrong's Hydraulic Machines, 950 

FRICTIONAL RESISTANCES. 

Steam Engines, 951 

Tools: — Shearing Machines — Plate-bending Machines — Circular Saws, '951 

VTork of Ordinary Cutting Tools, in Metal, 952 

Screw-cutting Machines — Wood-cutting Machines — Grindstones, . 954 

Colliery Winding Engines, 956 

Waggons in Coal Pits, 956 

Machinery of Flax Mills: — ^M. Comut*s Experiments, 957 

Hoise-|x>wer Required, 959 

Machinery of Woollen Mills:— Dr. Hartig's Experiments, . . . , . 959 

Machinery for the Conveyance of Grain, 960 

Traction on Common Roads: — M. Dupuit's Experiments — M. Debauve's De- 
ductions — M. Tresca's Experiments, 961 

Carts and Waggons on Roads and on Fields, 962 

Resistance on Railways, 965 

Resistance on Street Tramways, 966 

APPENDIX. 

Dr. Siemens' Water Pyrometer, 967 

Atmospheric Hammers, 967 

Bernays* Centrifugal Pumps, 968 

Steam-Vacuum Pump, . 969 

Index, . - .'.... 971 



AUTHORITIES CONSULTED OR QUOTED. 



American, United, Railway Master Car- 

Builders* Association, Standard Sizes of 

Bolts and Nuts by, 663. 
American Society of Civil Engineers, Journal 

of: — Mr. J. F. Flagg, on Steam-vacuum 

Pamps, 969. 
Anderson. Dr., on the Strength of Cast Iron, 

555- 

Anderson, William, on Heating Water by 

Steam, 465. 466, 468 ; Translation of Cher- 

noffs Paper on Steel, 616. 
Annales dts Mines: — M. Krest, on the Slip 

of Belts, 742. 
Annales des Fonts et Chaussies: — M. Him's 

Rope Transmitter of Power, 754. 
Annales du Ginie Civile: — M. Paul Havrez, 

on Heating Surface of Locomotives, 803. 
A mnals of Philosophy : — Mr. Dunlop, on Tor- 
sional Strength of Cast Iron, 565. 
Annuaire det Association des Inginieurs sort is 

de r^coU de Zi^/.— Rivelted Joints, 641. 

Armengaud, French Standard Bolts and 

Nuls, by, 683. 
Aimstrong. Sir Wm., on Evaporative Power 

of Coals, 785; his Hydraulic Machinery, 

950- 
Arson, Anemometer by. 892. 

A^by & Co., Work of Steam in Portable 

Engine by, 883. 

Audenet, on Surface-Condensers, 475. 



B 



BaJcer. B., on the Strength of Beams, 512; 

of Oak. 544, 549; of Columns, 645, 646; 

of RaQs, 662, 666. 
Barlow. Peter, on Strength of Timber, 547 ; 

of Cast Iron, 561 ; of Wrought Iron, 567, 

588, 590 ; of Iron Wire, 586. 
Bariow. W. H,. on the " Resistance of Flex- 
ure." 507. 
Bamaby. Mr., on Strength of Punched Steel 

Plate, 642. 
Barrow Hematite Steel Company, Strength 

of Steel made by, 594. 618, 619, 620, 621. 
Batcman, J. F., on Flow of Water through 

Submerged Openings. 930; his Cast-Iron 

Pipes. 934- 



Baudrimont, on Strength of Metallic Wires. 
628. 

Beardmore, on the Work of Horses, 720 ; on 
Limits of Velocity at the Bottom of a 
Channel, 934. 

Beaufoy, Colonel, on Resistance of Air, 897. 

Bell, J. Lothian, on the Heat in Blast Fur- 
naces, 498. 

Berkley, George, on the Strength of Cast- 
iron Beams. 647-650. 

Berkley, J., Specific Gravity of Indian Woods, 
by, 209. 

Bemays, Joseph, on Centrifugal Pumps, 968. 

Bertram, W., on Rivetted Joints, 634-637. 

Borsig, Herr, Strength of Wrought- Iron 
Plates, 586. 

Box, Thomas, on the Load on Journals, 766 ; 
Thickness of Gas Pipes, by, 936. 

Boyden, Outflow Turbine by, 940. 

Bradford, W. A., on Otto and Langen's Gas- 
Engine. 924. 

Bramwell. F. J., on Heating Water by Steam, 
467, 468: on the Strength of Cast Iron, 
556 ; on Portable Steam Engines, 801, 883, 
886 ; on the Expansive Working of Steam, 
889. 

Brereton, R. P., on Strength of Timber Piles, 
646. 

Briggs, Blowing Engine by, 927. 

British Associatiom, Transactions of : — F. W. 
Shields, on Strength of Cast-iron Columns. 

645- 
Brown & May, Work of Steam in Portable 

Engine by, 882. 
Brownlee, J., on Saturated Steam, 382; on 

the Outflow of Steam, 893 ; Flow of Water 

through a Submerged Nozzle, 931. 
Bruce, G. B., on the Work of a Labourer, 

719. 
Brunei, on the Strength of Rivetted Joints, 

638 ; and of Bolts and Nuts, 680. 
Buchanan, W. M., on Saturated Steam, 379. 
Buckle. W., on Fans, 924. 
Buel, R. H., on the Slip of Belts, 742. 
Bulletin de la Sociiti Industrielle de Mul- 

house: — M. Leloutre on Steam Engines, 

886. 
Burnat, on Condensation of Steam in Pipes, 

472. 474- 
Bury, Wm., on Strength of Flat Stayed Sur- 
faces, 686. 

<52 



XXll 



AUTHORITIES CONSULTED OR QUOTED. 



Cameron, Dr. , Analysis of Peat by, 454. 
Chari^-Marsaines, on Flemish Horses, 964. 
Chenot Ain^, Atmospheric Hammer by, 967. 
Chemoff, on Steel, 616. 
Chevandier, on Composition of Wood, 440; 
on its Weight and Bulk, 442, 443. 

Chevandier & Wenheim, on Strength of Tim- 

• ber, 538, 545, 546, 549. 

Clark, D. t^., on Proi>erties of Saturated 
Steam, 387; on Locomotive Boilers, 798; 
on the Work of Steam. 879, 880^ 884; on 
Resistance on Railways, 965. 

Clark, Edwin, on the Strength of Beams, 51a; 
of Red Pine, 543, 544, 549 ; of Cast Iron, 
562 ; of Bar Iron, 570, 588, 590, 623. 

Clark, Latimer, on Wire Gauges, 130. 

Claudel on Fuels and Woods, by, 207, 211, 
212 } tints of Heated Iron, 328 ; on Heating 
Factories, 486; on Heating Rooms, 488, 
489; on Belts, 743, 746; on Blowing En- 
gines, 927; on Pumps, 944. 

Clement, on Transmission of Heat, 462, 468 ; 
on Condensation of Steam in Pipes, 474 ; 
on Drying Stuffs, 496 ; on the Heat to Melt 
Iron, 497. 

Cochrane, J., on Strength of Perforated Bar 
Iron, 633. 

Cockerill, John, Blowing Engines by, 927. 

Colliery Guardian : — Mr. Mackworth on Ven- 
tilation of Mines. 480. 

Conservatoire cUs Arts et Metiers, Annates 
du: — Hot-Air Engines by Laubereau. and 
by Belou, 917-9x9 ; Gas-Engines by Lenoir, 
920; by Hugon, 921; by Otto & Langen, 

923- 
Cooper, J. H., on Very Wide Belts, 747, 749. 

Comet, on the Work of a Labourer in France, 
720. 

Comut, E., on Mill-Shafting, 766 ; on Machin- 
ery of Flax-Mills, 957 ; on Flow of Air in 
Pipes, 896. 

Cotterill. J. H., on Work of Compression of 
Air, 903. 

Cowper, E. A., Compound Engine by, 889. 

Craddock, Thomas, on Cooling through 
Plates, 461. 

Crighton & Co. , on Drying Grain, 496. 

Crookewitt, on Specific Gravities of Alloys, 

2CO. 

Crossley, F. W., on Otto & Langen's Gas- 

Engines, 923. 
Cubitt, Mr., on Strength of Cast-Iron Beams, 

649. 

D 

DagUsh, G. H., on Resistance of Colliery 
 Winding Engines, 956. 
Dalton, Dr., on " Spontaneous " Evaporation 
of Water, 491. 



Daniel, W., on Ventilation of Mines, 925. 
Dan vers, F. C, on Coal Economy, 4x0. 
Darcy, on Cooling Hot Water in Pipes, 469. 
D'Aubuisson, on Flow of Compressed Air, 
896 ; on Hydraulic Rams, 949. 

Davey, Paxman,. & Co., Work of Steam in 

Portable Engine by, 883. 
Davies. Thomas, on Strength of Rivetted 

Joists. 658. 
Davison, R., on Resistance of Shafting, 

766; Duty of Pumps by, 944; on Resist- 
ance of Grain Machinery, 961. 
Day, Summers, & Co., Work of Steam in 

Marine Engines by, 882. 
Debauve, on Resistance on Common Roads, 

961. 
Delabtehe & Playfair, on British and Foreign 

Coals, 206, 413, 416, 770. 
Despretz, on Conducting Powers of Bodies, 

331- 
Deville, Sainte-Claire, on Composition of 

Petroleum and other Oils, 456, 457. 
Dewrance, John, on the Heating Surface of a 

Locomotive, 803. 
Donkin, Bryan, & Co., Work of Steam in 

Stationary Engines by, 882. 
Downing, on Flow of Water in Pipes, 933. 

934- 
Dunlop, on Strength of Cast Iron, 565. 

Dupuit, on Resistance on Common Roads, 

961. 

Durie, James, on Rope-Gearing, 753. 

Duvoir, Ren^, Drying House by, 495. 



Eastons & Anderson, on Portable Steam 
Engines, "Box ; on Rider's Hot-Air Engine, 
9x7; on Resistance 'of Waggons. 962. 

Elder, John, & Co., on the Strength of Boilers. 
638, 693 ; Work of Steam in Marine Engine 
by, 882. 

Emery, on American Marine Engines, 884. 

Engineer, ZA*?.— Crighton & Co. on Drying 
Grain, 496 ; Mr. W. S. Hall on the Strength 
of Rivetted Joints, 641 ; Messrs. Woods & 
Dewrance on Locomotive Boilers, 803 ; Mr. 
C. L. Hett on Hydraulic Rams, 949. 

Engineering: — on Heating Water by Steam, 
464; on Cooling Wort, 470, 471; Mr. B. 
G. Nichol on Surface Condensation, 476; 
Mr. G. Graham Smith on Strength of 
Timber, 544; Factor of Safety for Wrought 
Iron, by Roebling, 625 ; Mr. W. S. Hall 
on the Strength of Rivetted Joints, 64X ; 
Mr. John Mason on Strength of Untanned 
Leather Belts, 680; Mr. Phillips on Strength 
of Flat Plates, 686; Mr. Bury on the Strength 
of Flat Stayed Surfaces, 686 ; Messrs. John 
Elder & Co. on the Strength of Boilers, 
638, 693 ; Mr. J. Durie on Rope Gearing, 



^ 



AUTHORITIES CONSULTED OR QUOTED. 



XXlli 



753; Dr. Martig- on Resistance of Tools, 

951; Resistance of Waggons, by Messrs. 

Eastons & Anderson, 962. 
Emglisk Mechanic : — Mr. W. A. Bradford on 

Olto & Langen's Gas-Engine. 924. 
E^Tard« A. on the Work of Animals, 720. 



Fagersta Steel Works, Strength of Steel made 
at, 604, 618, 619. 690, 621. 

Faiifaaim, Sir Williani, on Hot-Blast Iron, 
556; on the Strength of Cast Iron, 557; on 
the Strength of Wrought Iron, 567-569; 
of Rivetted Joints, 633 ; of Screwed Stay- 
Bolts and Flat Stayed Plates. 685 ; on the 
Proportions of Spur Wheels, 729, 734, 737 ; 
on the Load on Journals. 766, 767; on 
Water Wheels, 938. 

Fairbaim & Tate, on the Expansion of Steam, 

383- 
Fairweather, James C, on Resistance of Air, 

897- 
Faraday, Dr., on the Liquefaction of Gases, 

372. 
Favre & Silbermann, on the Heating Powers 

of Combustibles, 404. 
Field. Joshua, on the Work of Labourers, 719. 
Fincham, on Strength of Timber, 542, 543, 

549- 

Flagg, J. F., on Steam-vacuum Pumps, 969. 

Fletcher, L. E., on the Strength of a Boiler, 
638. 693 ; his Reports. 696 ; his Report on 
Bofler and Smoke Prevention Trials, 771- 
784. 

Yowke, Captain, on Colonial Woods. 209. 

Fowler, G., on Resistance of Waggons in 
Coal Hts, 956. 

Fowler, John, Strength of Steel Rails de- 
signed by, 666, 670. 

Fowler, J., & Co., Compressed-air Machinery 
by, 916. 

Fox. Head, & Co., on Condensation of Steam 
in a Boiler, 475. 

Francis, J. B., on a Swain Turbine, 943. 

Franilin Institute, Journal of: — the Shear- 
ing Resistance of Bar Iron, by Chief 
Engineer W. H. Shock, 588; Mr. R. H. 
Bud on Belts, 742 ; Mr. H. R. Towne on 
Belts, 742, 745; Mr. J. H. Cooper on 
Bells, 747: Mr. S. Webber on Mill Shaft- 
ing. 763, 764; Mr. Emery on American 
Marine Engines, 884; Mr. Briggs on 
Blowing Engines, 927; Mr. J. B. Francis 
on a Swain Turbine, 943; Mr. E. D. 
Leavitt's Pumping Engines. 948. 



Gammelbo & Co., Hammered Bars made by, 
Strength of, 581. 



GaudiUot; on Heating Apparatus, 486. 
Gay-Lussac. on Cold by Evaporation, 376. 
Glynn, Mr.| on Strength of Ropes, 673 ; on 

the Work of a Labourer, 718. 
Gooch, Sir Daniel, on Consumption of Water 

by the "Great Britain" Locomotive. 884. 
Gordon. L. D. B., on Strength of Colimms, 

645- 
Graham, John, on Heating Water, 461 ; on 

Heating Surface, 802. 
Grant, on Strength of Cements, &c., 630. 
Greaves, on Pumping Engines, 948. 
Grouvelle, on Condensation of Steam in Pipes, 

474; on Heating Factories, 486. 487. 



H 



Hackney, W., on Anthracitic Cbke, 432. 

Haines, R.. on Indian Coals, 423. 

Hall, W. S., on the Strength of Rivetted 

Joints, 64X. 
Harcourt, Vernon, on Analysis of Coal-Gas, 

458. 

Harmegnies, Dumont, & Co., on French Wire 
Ropes, 677. 

Hartig, Dr., on Driving Belts, 743; on Re- 
sistance ' of Tools, 951 ; on Resistance of 
Machinery of Woollen Mills, 959. 

Havrez, P.. on Heating Water by Steam, 
464, 468; on Heating Surface of Loco- 
motives, 803. 

Hawksley, Thomas, on Flow of Air through 
Pipes, 894 ; on Velocity of Air in Up-cast 
Shaft, 897 ; on Flow of Water in Pipes. 
933 ; on Thickness of Water Pipes, 935. 

Hett, C. L., on Hydraulic Rams, 949. 

Hick, John, M.P., on Friction of Leather 
ColUurs, 950. 

Him, on Work of Expanded Steam in Sta- 
tionary Engines, 886. 

Hodgkinson, on the Strength of Cast Iron, 

553-555. 558, 559. 563. 564; of Columns, 

643, 646; of Cast-iron Flanged Beams, 

647-650. 
Holtzapffel, his Wire-Gauges, 131, 13a, 134. 
Hood, on Warming and Ventilation, 477-485. 
Hopkinson, on the Performance of a Corliss 

Engine, 88 z. 

Hunt, R., on Combustion of Coal, 770. 

Hutton, Dr., Law of Resistance of Air by« 
897. 

JLa 

Institute of Naval Architects, Transactions 
of M^.— Strength of Rivet Joints of Sted 
Plates, 642. 

Institution of Civil Engineers, Proceedings 
of:— Mr. Wm. Anderson on Heating Water 
by Steam, 465; M. Bumat on Condensation 
of Steam in Pipes. 472; Dr. Pole on Spon- 
taneous Evaporation, 493; Regenerative 
Hot-Blast Stoves. 556: Mr. Bramwell on 



XXIV 



AUTHORITIES CONSULTED OR QUOTED. 



Strength of Cast Iron, 556; Mr. Grant on | 
the Strength of Cements, &c., 630; Mr. J. 
Cochrane on the Strength of Punched Bar 
Iron, 633; Mr. R. Price Williams on 
Strength of Rails, 662; Mr. J. T. Smith 
on the Strength of Bessemer Steel Rails, 
664; Mr. R. Davison on Resistance of 
Shafting, 766 ; Evaporative Performance of 
Steam Boilers in France, 796 ; Composition 
of Coals and Lignites, 797; M. Paul Havrez 
on Heating Surface of Locomotives, 803; 
Mr. Emery on American Marine Engines, 
884; Mr. Hawksley on Flow of Air through 
Pipes, 894 ; and on Velocity of Air in Up- 
cast Shaft, 897 ; M. Piccard on the Work 
of Compressed Air, 911 ; Mr. J. B. Francis' 
trial of a Swain Turbine, 943; Mr. R. 
Davison on Duty of Pumps, 944 ; Hon. R. 

C. Parsons on Centrifugal Pumps, 947 ; Mr. 
Henry Robinson on Armstrong's Hydraulic 
Machines, 950. 

Institution of Engineers and Ship-Buiiders in 
Scotland, Transactions of the: — on Strength 
of Helical Springs. 672 ; Report on Safety 
Valves, 893; Mr. J. Brownlee's Experi- 
ments on Flow of Water, 931. 

Institution of Mechanical Engineers, Pro- 
ceedings of: — Mr. C. Little on the Shearing 
and Punching Strength of Wrought Iron, 
587 ; Mr. Vickers on the Strength of Steel, 
621; Mr. W. R. Browne's paper on Rivetted 
Joints, 637; Mr. Robertson on Grooved 
Frictional Gearing, 741 ; Mr. H. M. Mor- 
rison on Him's Rope Transmitter, 755; 
Mr. Ramsbottom on Cotton- Rope Trans- 
mitter, 755; Mr. Westmacolt and Mr. B. 
Walker on Resistance of Shafting, 766 ; Mr. 

D. K. Clark on the Expansive Working of 
Steam in Locomotives, 879. 880; Data of 
the Practical Performance of Steam, 880; 
Mr. F. J. Bramwell on Economy of Fuel 
in Steam Navigation, 889 ; Compressed-Air 
Machinery by Messrs. John Fowler & Co., 
916 ; Wenham's Hot-Air Engine, 919 ; Mr, 
F. W. Crossley on Otto and Langen's Gas- 
Engine, 923; Mr. Buckle on Fans, 924; 
Mr.J.S.E. Swindell on Ventilation of Mines, 
925; Mr. W. Danielon Ventilation of Mines, 
925 ; Mr. A. C. Hill on Blowing Engines, 
927 ; Mr. J. F. Bateman's Experiments on 
Flow of Water, 930 ; Mr. David Thomson 
on Pumping Engines, 948; Mr. G. H. 
Daglish on Winding Engines. 956 ; Mr. G. 
Fowler on Resistance of Waggons in Coal 
Pits, 956 ; Mr. Westmacott on Com- Ware- 
housing Machinery, 961. 

Iron and Steel Institute, Journal of the: — 
Mr. J. Lothian Bell on the Cleveland Blast 
Furnaces, 498. 

Isherwood, Trials of Evaporative Performance 
of a Marine Boiler, 795. 



J 

James, Captain, on the Strength of Cast Iron, 

555- 
Jardine, Mr., on the Strength of Lead Pipes, 

696. 
Johnson, Professor W. R., on American Coals, 

418, 770. 791-795- 
Joule, Dr., Mechanical Equivalent of Heat, 
by, 332. 

K 

Kane, Sir Robert, on Peat, 453. 

Kennedy, Colonel J. P., on Weight and 
Volume of Goods carried on Railways, 213. 

Kirkaldy, David, on Compressive Strength 
of Timber, 546, 547, 647 ; on the Tensile 
Strength of Wrought Iron and Steel, 571- 
578 ; of Swedish Hammered Bars. 581, 590; 
of Krupp and of Yorkshire Iron Plates, 
583-586 ; of Borsig's Iron Plates, 586; Ten- 
sile Strength of Bar Steel 593, 594 ; of He- 
matite Steel, 594 ; of Krupp Steel, 595 ; of 
Steel Bars, for the Steel Committee, 597- 
600; of Fagersta Steel, 604-611 ; of Siemens- 
Steel Plates and Tyres, 612-614 ; on Shear- 
ing Strength of Steel. 617; on Strength of 
Phosphor-Bronre, 628, 629; of Wires, 629; 
of Rolled Wrought-iron Joists, 654; of Rails. 
662, 663, 666-668; of Ropes, 674; of Belt- 
ing, 680; of Plates of a Marine Boiler, 694. 

Krest, on the Slip of Belts, 742. 

Krupp, Herr, Strength of Wrought-Iron Plates 
made by, 583 ; of his Cast Steel, 595. 618- 
62T. 

L 

Landore Siemens-Steel Company, Strength of 
Steel Plates and Tyres made by, 6i*-6i4. 

Laslett. Thomas, on the Strength of Timber, 
538-542. 546, 548, 550. 647. 

Leavitt, E. D., Pumping Engines by. 948. 

Legrand, on Boiling Points, 370. 

I^igh, Evan, on Belting, 746. 

Ldoutre, on M. Him's Experiments on Work 
of Steam, 886. 

Leplay. on Moisture in Wood, 439; on 
Drying Wood, 496. 

Literary and Philosophical Society of Man- 
chester, Memoirs of: — Dr. Dalton on 
" Sipontaneous " Evaporation, 491; Mr. 
John Graham on Heating Surface, 802. 

Little, C, on the Shearing and Punching 
Strength of Wrought Iron, 587. 

Lloyd, Thomas, on the Strength of Bar Iron. 

569. 570. 
London Association of Foremen Engineers, 

Proceedings of: — Mr. David Thomson on 

Expansive Work of Steam, 822. 

Longridge, J. A., on Combustion and Evapn 
orative Power of Coals, 770, 785. 

Longsdon, Mr., on Strength of Krupp Steel, 

595- 



AUTHORITIES CONSULTED OR QUOTED. 



XXV 



M 



MacCon, on the Strength of Rivetted Joints, 

641. 
Mackintosh. Charles, Weight of Belt-Pulleys 

by. 75a. 
Mackworth. H., on Ventilation of Mines, 

479- 
Madure. H. H., on Strength of Timber, 542, 

543. 549- 
Macneil, Sir John, on Resistance on Common 

Roads; 964. 
Mahan, Lieutenant F. A., on Outward-Flow 

Tuibines, 941. 
Mallaid, on Compressed-Air Machines, 90a ; 

on Compressed Air, 907, 91a. 
Mallet, R.. Strength of Buckled Iron Plates 

by, 660. 
Marshall Sons, & Co., Work of Steam in 

Portable Engine by, 883. 
Mason, John, Strength of Untanned I.^eather 

Belts by, 680. 
M'Donndl, A, on Composition of Peat, 454. 
Menelaus, on Portable Steam Engines, 801. 
Miller, T. W., Trials of Coals by, 790. 
Miner & Taplin. Trials of Coals by, 787. 
Montgoilfier, on Drying by Forced Currents, 

494- 

Monthly Reports to the Manchester Steam- 
Users' Association: — Mr. L. E. Fletcher's 
Data. 696. 

Morin. on Transverse Strength of Timber, 
537; on the Friction of Journals, 722 ; and 1 
of Solid Bodies, 733 ; on Leather Belts, 
743-745; *5^ Breast Wheels, 938; on a 
Foomeyron Turbine. 940; on Centrifugal 
Pomps, 946. 

Morrison. H. M., on M. Him's Rope Trans- 
mitter, 755- 

Morton, Francis, & Co., Weight of Jron 
fleets by, 245 ; Strength of Cable Fencing 
Stands by, 676. 

Moser. Strength of Beams tested for, 654. 

Muspratt. Dr.. Analyses of Coke by, 433. 



N 



Naper, James R., on Transmission of Heat, 

460 ; on Drying Stuffs, 496. 
Nao. on Moisture in Charcoal, 451. 
Xen-all. R. S., & Co., Strength of Hemp and 

Wire Ropes by, 674. 
Nichol, B. G., on Condensation of Steam in 

Pipes and Tubes, 474, 476. 
Xjcon & Lynn. Trials of Coals by, 784. 
Norris & Co.. Strength of Leather Belts by, 

68a 
North British Rubber Company, Driving Belts 

by. 730- 

Af'orth of England Mining Institute, Transac- 
tions e^.— Rivetted Joints, 588. 



O 



Oldham, Dr., on Indian Coals, 424. 
Ott, Karl Von, on Strength of Ropes, 674, 
679. 

P 

Parsons, on Strength of Oak Trenails, 551. 
Parsons, Hon. R. C, on Centrifugal Pumps, 

947- 

Payen. on Explosive Mixture of Gas and Air, 
921. 

Pearce, W. A., on Rope Gearing, 754. 

Peclet, on Radiation of Heat, 329 ; on French 
Coals. 420 ; on Coke, 431 ; on Moisture in 
Tan, 455; on Transmission of Heat, 459, 
46a, 463, 468; on Condensing Power of 
Air and Water, 475 ; on Ventilation, 477 ; 
on Heating Apparatus, 488, 489 ; on Drying 
by Air Currents, 494 ; on a Drying House, 
495 ; on Cupola Furnaces, 497. 

Penot, on Drying Houses, 496. 

Penrose & Richards, their Anthracitic Coke, 

432- 
Perkins, Heating Apparatus by, 486. 

Perkins, Jacob, Invention of the Ice-Making 
Machine by, 373. 

Person, on the Latent Heat of Fusion, 367. 

Phillips, on Strength of Flat Plates, 686. 

Piccard, on Work of Compressed Air, 911. 

Poir^ on Friction on Rails by, 724. 

Pole, Dr., on Spontaneous Evaporation, 493 ; 
on the Strength of Steel Wire, 617. 

Poncelet, on Water Wheels, 938. 

Forte/euille de John Cockerill: — Blowing 
Engines, 927. 

Porter, C. T., on Expansion of Steam, 886. 

Pouillet, on Luminosity at High Temper- 
atures, 328. 

R 

Radford, R Heber, Weight of Belt-Pulleys 

by,. 751. 752. 
Ramsbottom, J., on Cotton- Rope Transmitter, 

755- 
Rankine, Dr. , on Expansion of Water, 340 ; 

on the Melting Point of Ice. 364; on 

Transmission of Heat, 461 ; on Shearing 

Strength of Oak Trenails, 551 ; and of 

Cast Iron, 561 ; Factors of Safety, 625, 626 ; 

on Stresses in Roofs, 715, 717 ; on Load on 

Working Surfaces. 767. 

Reading Engine Works Co., Work of Steam 
in Portable Engine by, 883. 

R^clus, Specific Gravity of Sea Water by, 126. 

Regnault, Air Thermometer by, 325 ; on the 
Expansion of Air, 344 ; on Specific Heat 
of Metals, 353; and Gases, 359; Boiling 
Points of Vapours, 371 ; on Steam, 378, 
379, 383, 384 ; on the Mixture of Gases and 



XXVI 



AUTHORITIES CONSULTED OR QUOTED. 



Vapours, 392 ; on French Coals, 420, 421 ; 
on Lignite and Asphalte, 436. 
Reilly, Calcott, on the Varieties of Stress, 

500- 
Rennie, on the Work of Horses, 720: 

Reime IndustrielU: — Atmospheric Hammer 

by M. Chenot Ain^, 967. 
Reynolds, Dr., on Peat, 454. 
Richardson, Dr., on Coals, 412 ; on Coke, 

433; Report on Evaporative Power of Coals, 

785- 
Robertson, James, on Grooved Frictional 

Gearing, 741. 
Robinson, Henry, on Armstrong's Hydraulic 

Machines, 950. 
Roebling, on the Strength of Iron Wire, 587 ; 

and of Steel Wire, 617 ; Factor of Safety for 

Iron, 625 ; on the Strength of Wire Rope 

and Hemp Rope, 676. 
Ross, Owen C. D., on Coal Gas, 457. 
Rouget de Lisle, on Drying StufTs, 496. 
Royal Society of Edinburgh, Proceedings of: — 

Mr. Fairweather on Resistance of Air, 897. 
Royer, on Drying Houses, 496; on Drying 

Stuffs, 496. 
Russell & Sons, J., on the Strength of 

Wrought-Iron Tubes, 692, 693. 
Ryland Brothers, Warrington Wire Gauge by, 

133. 247- 



Sauvage, on Charcoal, 447, 449, 452. 
Scheurer-Kestner & Meunier - DoUfus, on 
French and other Coals, and Lignites, 422, 

797- 
Sharp, Henry, on Rivetted Joints of Steel 

Plates, 642. 

Shields, F. W., on Cast-Iron Columns, 645. 

Shock. Chief Engineer W. H., on Shearing 
Strength of Bar Iron, 587. 

Siemens, Dr. C. W., on Isolated Steam, 383; 
on the Consumption of Fud in Metallurgical 
Furnaces, 497; on the Strength of Hot- 
Blast Iron, 556 ; on Hot-Air Engines, 920 ; 
his Water Pyrometer, 967. 

Simms, F. W., on the Work of Horses, 720. 

Smeaton. on the Power of Labourers. 718. 

Smith, G. Graham, on Strength of Timber, 

543. 544. 549- 
Smith, J. T., on Punching Resistance of Steel, 

617 ; on the Strength of Rails, 664. 

Snelus, G. J., Analysis of Welsh Coal by, 413. 

SocUU IndustrielU de Mulhouse: — on Steam 
Boilers, 796. 

Soctiti IndustrielU Minerale, Bulletin de 
la: — M. Comut on Compressed-Aif Machi- 
nery, 896 ; M. Mallard on Compressed-Air 
Machines, 902. 

Sociiti des InginUurs Civils, Comptes Rendus 
de la: — Anemometer by M. Arson. 892. 

SociH6 Vaudoise des Inginieurs et des Archi- 



tecies, Bulletin de la: — M. Piccard on 

Compressed Air, 911. 
Society of Arts, Committee of, on Resistance 

on Common Roads, 963. 
Society of Arts, Journal of: — on Resistance 

on Common Roads, 963. 
Spill, Strength of Belting by, 680. 

Steel Committee of Civil Engineers, on the 
Strength of Wrought Iron, 579, 580 ; and of 
Steel, 596-603, 

Stephenson, Robert, on the Strength of Cast 
Iron, 555, 561. 

Stoney, on Stress in a Curved Flange, 525; 
on Sectional Area of a Continuous Web, 
526; on Shearing Strength of Cast Iron, 
561 ; his Factors of Safety, 625 ; on the Re- 
sistance of Columns, 643, 645, 646; on 
Stresses in Roofs, 715. 

Sullivan, Dr., on Peat, 207. 

SutclifTe, on Condensation of Steam in the 
Cylinder, 880. 

Swindell, J. S. E,, on Ventilation of Mines, 

935- 
Sylvester, Cockle Stove by, 488. 



Tangye, J., on the Compressive Resistance of 

Wrought Iron, 582. 
Tasker, Work of Steam in Portable Engine 

by, 883. 
Telford, Thomas, on the Strength of Wrought 

Iron, 567 ; and of Iron Wire, 586. 
Thomas & Laurens, on Brown Charcoal, 449; 

on Heating by Steam, 463, 468. 
Thomson, David, on Expansive Action of 

Steam, 822, 882; on Centrifugal Pumps, 

946 ; Duty of Pumping Engines, 948. 
Thomson, Professor James, Vortex Wheel 

by. 943- 
Thurston, on the Strength of Iron Wire, 587. 
Thwaites & Carbutt, on Root's Blower, 928. 
Towne, H. R., on Leather Belts, 679, 742, 

745. 748-750- 
Tredgold, Weight and Volume of Various 

Substances by, 213 ; on Cooling Hot Water, 

469; on Cooling of Steam in Pipes, 472, 

474 ; on the Work of a Horse, 720. 

Tresca, on Laubereau's Hot-Air Engine, 
917; on Gas-Engines, 920. 921, 923; on 
Pumps, 945, 946 ; on Resistance of Tram- 
way Omnibus, 961. 

Turner, Work of Steam in Portable Engine 
by, 883. 

Tweddell, R. H., on Shafting, 763. 



U 



Umber, on M. Him's Wire Ropes. 754. 
Unwin, on Strength of Columns, 645. 
Ure, Specific Gravity of Alloys by, 20a 



AUTHORITIES CONSULTED OR QUOTED. 



XXVll 



Vidcers, T. K., on the Strength of Steel, 621, 

622. 
\'»lette. on Wood, 439, 441, 44a, 445; on 

Charcoal, 446-448. 450, 451 



W 



Wade, Major, on the Strength of Cast Iron, 

557- 
Walker, B., on Resistance of Shafting, 766. 
Walker. John, on the Work of Labourers, 

718. 
Webb, F. "NV.. on the Strength of Steel, 614, 

621. 
Vfdtibcr, S.. on Mill Shafting. 763. 764, 766. 
Westmacott, Percy, on Shafting, 766; on 

Com - Warehousing Machinery, 961; on 

Axuibt i o og's Hydraulic Machines, 950. 
WhiteUw, James, Water Mill by, 939. 



Whitworth, Sir Joseph, Standard Wife-Gauge 
by, 133, 134; Strength of his Fluid-Com- 
pressed Steel and of Iron, 614, 615; on 
Resistance of Steel and Iron to Explosive 
Force, 622; his System of Standard Sizes 
of Bolts and Nuts, 681 ; Standard Pitches 
of Screwed-Iron Piping, 683. 

Wiesbach, Coefl&cients for Flow of Water, 
892. 

Williams, R; Price, on the Transverse Strength 
of Rails, 662, 664. 

Williams, Foster, & Co., Weight of Sheet 
Copper by, 261. 

Wilson, A., on the Work of Bullocks, 720. 

Wilson, R., on Sttength of Perforated Iron 
Plates, 633. 

Wilson, Robert (Patricroft), on Teeth of 
Wheels, 732. 

Wood, J. & E., Work of Steam in Stationary 
Engine by, 882. 

Woods, E., and J. Dewrance, on the Efficiency 
of Heating Surface of a Locomotive, 803. 

Wright, J. G., on.Rivetted Joints, 637. 



A MANUAL 



OF 



RULES, TABLES, AND DATA 



FOR 



MECHANICAL ENGINEERS. 



GEOMETRICAL PROBLEMS. 



PROBLEMS ON STRAIGHT LINES. 

Problem I. — To bisect a straight 
iine^ or an arc of a circle^ Fig. i. — 
From the ends a, b, as centres, de- 
scribe arcs intersecting at c and d, 
and draw c d, which bisects the line, 
or the arc, at the point e or f. 

Problem IL — To draw a perpen- 
dicuiar to a straight line^ or a radial 
line to a circtilar arCy Fig. i. — Operate 



:c 



 ^ 

.4. 



:b : 




:d *• 



F^5. I. — Probs. L and II. 

as in the foregoing problem. The 
line CD is perpendicular to a b : the 
line c D is also radial to the arc a b. 

Problem IIL — To draw a perpen- 
dicular to a straight line, from a given 
point in that iinc,Yig. 2. — With any 
radius, from the given point a, in the 



line B c, cut the line at b and c; with 
a longer radius describe arcs from b 



Fig. 2.— Prob. III. 

and c, cutting each other at d, and 
draw the perpendicular d a. 

2d Method, Fig. 3. — ^rom any cen- 
tre F, above bc, describe a circle 
passing through the given point a, 




Fig. 3.— Prob. III. 3d method. 

and cutting the given line at d; draw 
D F, and produce it to cut the circle 
at e; and draw the perpendicular a e. 



GEOMETRICAL PROBLEMS 



3// Method^ Fig. 4. — From a de- 
scribe an arc eg, and from e, with 
the same radius, the arc a c, cutting 



/ 









'/"\ 



Fig. 4, — Prob. III. 3d method. 

the Other at c ; through c draw a Une 
E c D, and set off c d equal to c e ; 
and through d draw the perpendicu- 
lar AD. 

4//z Method, Fig. 5. — From the 
given point A set off a distance a e 

••c 



m 



^' 



» •• 



.. 4 



E' 



Fig. 5.— Prob. III. 4th method. 

equal to three pjrrts, by any scale; 
and on the centres a and e, With 
radii of four and five parts respec- 
tively, describe arcs intersecting at c. 
Draw the perpendicular a c. 

Note, — This method is most useful 
on very large scales, where straight 
edges are inapplicable. Any multi- 
ples of the numbers 3, 4, 5 may be 
taken with the same effect, as 6, 8, 
10, or 9, 12, 15. 

Problem IV. — To draw a perpen- 
diadar to a straigJit line from any 
point without it, Fig. 6. — From the 
point A, with a sufficient radius, cut 
the given line at rand g; and frt)m 
these points describe arcs cutting at 
e. Draw the perpendicular a e. 

Note. — If there be no room below 



the line, the intersection may be taken 
above the line; that is to say, be- 
tween the line and the given point. 




Fig. 6.-Prob. IV. 

2d Method, Fig. 7. — From any two 
points B, c, at some distance apart, 



F 

/ 



B 



Fig. 7.— Prob. IV. ad method. 

in the given line, and with the radii 
B A, c A, respectively, describe arcs 
cutting at a d. Draw the perpendi- 
cular A D. 

Problem V. — 72? draw a straight 
line parallel to a giveti line, at a giten 
distance apart. Fig. 8. — From the cen- 



c 


D 


,.- 


^- -... 










1- .... 

i 


• 



B 



Fig. 8.— Prob. V. 



tres A, B, in the given line, with the 
given distance as radius, describe arcs 
c, D ; and draw the parallel line c D 
touching the arcs. 

Problem VI. — To draw a parallel 
through a given point. Fig. 9. — With 
a radius equal to the distance Of the 



ON STRAIGHT LINES. 



given point c from the given line 
A B, describe the arc d from b, taken 



1 




.-^ ■--. 


A 

rig. 


9-- 


B 
-Prob. VI. 



considerably distant from c Draw 
the parallel through c to touch the 
arc D. 

2d Method^ Fig. lo. — From a, the 



— !p 



• 

I 

 
_t 



Fig. lo.— Prob. VI. ad method. 

given point, describe the arc f d, cut- 
ting the given line at f; from f, with 
the same radius, describe the arc e a, 
and set oflf f d equal to e a Draw 
the parallel through the points a, d. 

Note, Fig. II. — When a series of 
parallels are required perpendicular 
to a base line a b, they may be drawn, 
as in Fig. i, through points in the 
base line, set oflf at the required dis- 















t 

\ 
* 






•B 



Fig. XX.— Prob. VI. 

tamces apart This method is con- 
venient also where a succession of 
parallels are required to a given line, 



c D ; for the perpendicular a b may be 
drawn to it, and any number of par- 
allels may be drawn upon the per- 
pendicular. 

Problem VII. — To divide a straight 
line into a number of equal parts ^ Fig. 
12. — To divide the line a b into, say, 
five parts. From a and b draw par- 
allels A c, B D, on opposite sides. Set 
off any convenient distance four times 






>-' 



A^ 



X — % 



X 






-V 



^ 



Fig. 13.— Prob. VII. 

(one less than the given number) 
from A on AC, and from b on bd; 
join the first on a c to the fourth on 
B D, and so on. The lines so drawn 
divide a b as required. 

2d Method^ Fig. 13. — Draw the line 
A c at an angle from a, set off, say. 




h % 9 A 

Fig. 13.— Prob. VII. ad method. 

five equal parts; draw b 5, and draw 
parallels to it from the other points 
of division in a c. These parallels 
divide a b as required. 

Note. — By a similar process a line 
may be divided into a number of 
unequal parts; setting oflf divisions 
on A c, proportional by a scale to the 
required divisions, and drawing par- 
allels cutting A 6. 

Problem VIII. — Upon a straight 



GEOMETRICAL PROBLEMS 



line to draw an angle equal to a given 
angle. Fig. 14. — I^t a be the given 
angle, and fg the line. -With any 



radius, from the points a and f, de- 
scribe arcs D E, I H, cutting the sides 
of the angle a, and the line f g. Set 





Fig. 14.— Prob. VIII. 



ofl the arc i h equal to d e, and draw 
F H. The angle f is equal to a, as 
required. 

To draw angles of 60** and 30°, Fig. 
1 5. — From F, with any radius f i, de- 
scribe an arc i h ; and from i, with 
the same radius, cut the arc at h, and 



.H^. 




F K I 

Fig. 15.— Prob. VIII. 



draw F H to form the required angle 
I F H. Draw the perpendicular h k 
to the base line, to form the angle of 
30*" F H K. 

To draw an angle of 45**, Fig. 16. 
— Set off the distance f i, draw the 




i.r I 

Fig. 16.— Prob. VIII. 

perpendicular i h equal to i f, and 
join H F, to form the angle at f as re- 
quired. The angle at h is also 45**. 

Problem IX. — To bisect an angle, 
Fig. 17. — Let A c B be the angle; on 
the centre c cut the sides at a, b. On 
A and B, as centres, describe arcs 



cutting at d. Draw c d, dividing the 
angle into two equal parts. 




Fig. 17. — Prob. IX. 

Problem X. — To bisect the inclina- 
tion of two lines, of which the intersec- 
tion is ifiaccessible, Fig. 18. — Upon the 




B 
Fig. 18.— Prob. X. 



given lines cb, ch, at any points, 
draw perpendiculars e ^, g h, of equal 
lengths, and through f and g draw 
parallels to the respective lines, cut- 
ting at s; bisect the angle fsg, so 
formed, by the line s d, which divides 
equally the inclination of the given 
lines. 



ON STRAIGHT LINES AND CIRCLES. 



5 



PROBLEMS ON STRAIGHT LINES 
AND CIRCLES. 

Problem XI. — Through tuto given 
points to describe an arc of a circle with 
a given radius^ Fig. 19. — On the points 




Problem XIIL — To describe a cir- 
de passing through three given points^ 
Fig. 21. — Let A, B, c be the given 
points, and proceed as in last pro- 



Fig. 19.— Proh. XI. 

A and B as centres, with the given 
radios, describe arcs cutting at c; and 
from c, with the same radius, describe 
an arc a b as required. 

Problem XII. — To find the centre 
of a cirde^ or of an arc of a drcle, 
I St, for a circle, Fig. 20. — Draw the 




Fig. 21.— Prob. XII. XIII. 

blem to find the centre o, from which 
the circle may be described. 

Note, — ^This problem is variously 
useful: — in striking out the circular 
arches of bridges upon centerings, 
when the span and rise are given; 
describing shallow pans, or dished 




iM 



Fig. aa— Prob. XII. 

chord A b, bisect it by the perpendi- 
cular c D, bounded both ways by the 
circle; and bisect c d for the centre g. 
2d, for a circle or an arc, Fig. 21. 
— ^Select three points, a, b, c, in the 
circumference, well apart; with the 
same radius, describe 2C£t!& from these 
three points, cutting each other; and 
draw the two lines, d e, f g, through 
their intersections, according to Fig. i. 
The point o, where they cut, is the 
centre of the circle or arc. 



Fig. 82.— Prob. XIV. xst method. 

covers of vessels ; or finding the dia- 
meter of a fly-wheel or any other 
object of large diameter, when only 
a part of the circumference is ac- 
cessible. 

Problem XIV. — To describe a drcle 
passing through three given points when 
the centre is not available, 

1st Mdhod, Fig. 22. — From the 
extreme points a, b, as centres, de- 
scribe arcs AH, BG. Through the 
third point c, draw a e, b f, cutting 



GEOMETRICAL PROBLEMS 



the arcs. Divide a f and b e into 
any number of equal parts, and set 
off a series of equal parts of the same 
length on the upper portions of the 
arcs beyond the points e, f. Draw 
straight lines, b l, b m, &c., to the divi- 
sions in A f; and a i, a k, &c., to the 
divisions in e g ; the successive inter- 



sections N, o, &c., of these lines, are 
points in the circle required, between 
the given points a and c, which may 
be filled in accordingly: similarly the 
remaining part of the curve b c may 
be described. 

2d Methody Fig. 23. — Let a, d,b be 
the given points. Draw a b, a d, d b, 




Fig. 23. — Prob. XIV. 2d method. 



and ef parallel to a b. Divide d a 
into a number of equal parts at i, 2, 3, 
&c., and from d describe arcs through 
these points to meet ef. Divide the 
arc A e into the same number of equal 
parts, and draw straight lines from d 
to the points of division. The inter- 
sections of these lines successively 
with the arcs i, 2, 3, &c., are points 
in the circle which may be filled in as 
before. 

Note. — ^The second method is not 
perfectly exact, but is sufficiently near 
to exactness for arcs less than one- 
fourth of a circle. When the middle 
point is equally distant fi-om the ex- 
tremes, the vertical c d is the rise of 
the arc; and this problem is service- 
able for setting circular arcs of large 
radius, as for bridges of very great 




Fig. 24.— Prob. XV. 

span, when the centre is unavailable; 
and for the outlines of bridge-beams, 



and of beams and connecting-rods of 
steam-engines, and the like. 

Problem XV. — To draw a tangent 
to a circle from a given point in the 
circumference^ Fig. 24. — Through the 
given point a, draw the radial line 




— E 



Fig. 25. — Prob. XV. 2d method. 

A c, and the perpendicular f g is the 
tangent 

2d Method^ when the centre is not 
available, Fig. 25. — From a, set off 
equal segments a b, a d; join b d, and 
draw A e parallel to it for the tangent 

Problem XVI. — To draw tangents 
to a circle from a point without it. 




Fig. 36.— Prob. XVI. ist method. 

\5t Method^ Fig. 26. — Draw ac 
from the given point a to the centre 



ON STRAIGHT LINES AND CIRCLES. 



G; bisect it at d, and from the centre 
D, describe an arc through c, cutting 
the circle at e, f. Then a e, a f, are 
tangents. 

id Method^ Fig. 27. — ^From a, with 
the radius a c, describe an arc b c d, 
and from c, with a radius equal to the 




•.\i> 



Fig. vj. — Prob. XVI. 2d method. 

diameter of the circle, cut the arc at 
B, D ; join B c, CD, cutting the circle 
at E,F, and draw ae, af, the tan- 
gents. 

Note. — ^When a tangent is already 
drawn, the exact point of contact 
may be found by drawing a perpen- 
dicular to it from the centre. 

Problem XVIL — Between two in- 
clined lines to draw a series of circles 
touching these lines and touching ecuh 
ather^ Fig. 28. — Bisect the inclination 




Fig. a8.— Prob. XVII. 

of the given lines a b, c d by the line 
NO. From a point p in this line, 



draw the perpendicular p b to the line 
A B, and on p describe the circle b d 
touching the lines and cutting the 
centre line at e. From e draw e f 
perpendicular to the centre line, cut- 
ting A B at F, and from f describe an 
arc E G, cutting a b at g. Draw g h 
parallel to b p, giving h, the centre 
of the next circle, to be described 
with the radius h e, and so on for the 
next circle i n. 

Inversely, the largest circle may 
be described first, and the smaller 
ones in succession. 

Note, — This problem is of frequent 
use in scroll work. 

Problem y>N\\l,^^ Between two 
inclined lines to draw a circular seg- 
ment to fill the angle, and touching the 
lines, Fig. 29. — Bisect the inclination 




Fig. 99. -Prob. XVIII. 

of the lines a b, d e by the line f c, 
and draw the perpendicular a f d to 
define the limit within which the cir- 
cle is to be drawn. Bisect the angles 
a and D by lines cutting at c, and 
from c, with radius c f, draw the arc 
H F G as required. 

Problem XIX. — To describe a cir- 
cular arc joining two circles, and to 
touch one of tliem at a given point. Fig. 
30. — ^To join the circles a b, f g, by 
an arc touching one of them at f, 
draw the radius e f, and produce it 
both ways; set off fh equal to the 
radius ac of the other circle, join ch 



8 



GEOMETRICAL PROBLEMS 



and bisect it with the perpendicular 
L I, cutting E F at i. On the centre i, 




Fig. 30. — Prob. XIX. 

With radius i f, describe the arc f a as 
required. 



PROBLEMS ON CIRCLES AND 
RECTILINEAL FIGURES. 

Problem XX. — To construct a tri- 
angle on a given base, the sides being 
given, 

I St. An equilateral triangle, Fig. 31. 




Fig. 31.— Prob. XX. 

— On the ends of the given base, a, b, 
with A B as radius, describe arcs cut- 
ting at c, and draw a c, c b. 

2d. A triangle of unequal sides, 
Fig. 32. — On either end of the base 
A D, with the side b as radius, describe 
an arc ; and with the side c as radius, 
on the other end of the base as a 
centre, cut the arc at e. Join a e, d e. 

Note, — This construction may be 
used for finding the position of a 
point, c or e, at given distances from 



the ends of a base, not necessarily to 
form a triangle. 



A 




Fig. 32.-Prob. XX. 



Problem XXI. — To construct a 
square or a rectangle onagiveti straight 
line, 

I St. A square. Fig. 33. — On the 




Fig. 33. — Prob. XXI. 

ends A, b, as centres, with the line a b 
as radius, describe arcs cutting at c; 
on c, describe arcs cutting the others 
at D E ; and on d and e, cut these at 
F G. Draw A F, B G, and join the in- 
tersections H, L 

2d. A rectangle. Fig. 34. — On the 
base E F, draw the perpendiculars e h, 



K 



Fig. 34.-Prob. XXL 

F G, equal to the height of the rect- 
angle, and join g h. 

When the centre lines, a b, c d, 
Fig. 35, of a square or a rectangle are 
given, cutting at e. — Set off e f, eg, 



ON CIRCLES AND RECTILINEAL FIGURES. 



the half lengths of the figure, and e h, 
ET, the half heights. On the centres 
H, T, with a radius of half the length, 



•B 



1^ 



A-:?' 



B 






Fig- 35.— Prob. XXI. 

describe arcs; and, on the centres F, 
G, with a radius of half the height, cut 
these arcs at k, l, m, n. Join these 
intersections. 

Problem XXII. — 7b construct a 
parallelogram^ of which the sides and 
one of the angles are given, Fig. 36. — 



CiLLl 





B 



Fig. 36.— Prob. XXII. 

Ehaw the side d e equal to the given 
length A, and set off the other side 
D F equal to the other length b, form- 
ing the given angle c. From e, with 
D F as radius, describe an arc, and 
from F, with the radius d e, cut the 
arc at G. Draw f o, eg. 

Or, the remaining sides may be 
drawn as |)arallels to d e, d f. 

The formation of the angle d is 
readily done as indicated, by taking 
the straight length of the arc h i and 
CI as radius, and finding the inter- 
section L. 

Problem XXIII. — To describe a 
circle about a triangle, Fig. 37. — Bisect 
two sides a b, a c of the triangle at 
E, F, and from these points draw per- 
pendiculars cutting at k. On the 



centre k, with the radius K a, draw 
the circle a b c. 




Fig. 37.— Prob. XXI 1 1. 

Problem XXIV. — To inscribe a 
circle in a triangle. Fig. 38. — Bisect 
two of the angles a, c, of the triangle 
by lines cutting at d; from d draw a 
perpendicular d e to any side, and 
with D E as radius describe a circle. 

When the triangle is equilateral, 
the centre of the circle may be found 
by bisecting two of the sides, and 




Fig. 38.— Prob. XXIV. 

drawing perpendiculars as in the pre- 
vious problem. Or, draw a perpen- 
dicular from one of the angles to the 
opposite side, and from the side set 
off one-third of the perpendicular. 




Fig. 39.— Prob. XXV. 

Problem XXV. — To describe a 
circle about a square, and to inscribe 
a square in a circle, Fig. 39. 



10 



GEOMETRICAL PROBLEMS 



I St. To describe the circle. Draw 
the diagonals a b, c d of the square, 
cutting at e; on the centre e, with the 
radius e a, describe the circle. 

2d. To inscribe the square. — Draw 
the two diameters a b, c d at right 
angles, and join the points a, b, c, d 
to form the square. 

Note, — In the same way a circle 
may be described about a rectangle. 

Problem XXVL — To inscribe a 
circle in a square^ and to describe a 
square about a circle^ Fig. 40. 

ist. To inscribe the circle. — Draw 




Fig. 4o.-^Prob. XXVI. 

the diagonals a b, c d of the square, 
cutting at e; draw the perpendicular 
e F to one side, and with the radius 
E F describe the circle. 

2d. To describe the square. — Draw- 
two diameters a b, c d at right angles, 
and produce them; bisect the angle 
D E b at the centre by the diameter 
F G, and through f and g draw per- 




Fig. 41.— Prob. XXVII. 

pendiculars ac, bd, and join the 
points A D and b c, where they cut the 
diagonals, to complete the square. 



Problem XXVIL — To inscribe a 
pentagon in a circle^ Fig. 41. — Draw 
two diameters a c, b d at right angles, 
cutting at o; bisect a o at e, and from 
E, with radius e b, cut a c at f ; from b, 
with radius b f, cut the circumference 
at G, H, and with the same radius step 
round the circle to i and k; join the 
points so found to form the pentagon. 

Problem XXVIIL — To construct 
a hexagon upon a given straight line^ 
Fig. 42. — From a and b, the ends of 
the given line, describe arcs cutting 
at^; from^, with tlie radius^ a, de- 




 ---^ B 



Fig. 4a.-Prob. XXVIII. 



scribe a circle; with the same radius 
set off the arcs a g, g f, and b d, d e. 
Join the points so found to form the 
hexagon. 

Problem XXIX. -^72? inscribe a 
hexagon in a circle^ Fig. 43. — Draw a 
diameter AC B; from a and b as centres, 
with the radius of the circle a c, cut 
the circumference at d, e, f, g; and 
draw AD, D E, &c. to form the hexagon. 




o -^9 

Fig. 43.-Prob. XXIX 



The points d, e, &c., may also be 
found by stepping the radius six 
times round the circle. 



"N 



ON CIRCLES AND RECTILINEAL FIGURES. 



II 



Problem XXX. — To describe a hex- 
a^m aJbaut a circle^ Fig. 44. — Draw a 




a *— . 



F7g. 44. — Prob. XXX. 



diameter adb, and with the radius 
A D, on the centre a, cut the circum- 
ference at c; join ac, and bisect it 
with the radius d e; through £ draw 
the parallel f g cutting the diameter 
at F, and with the radius d f describe 
the circle f h. Within this circle de- 
scribe a hexagon by the preceding 
problem; it touches the given circle. 

Problem XXXI, — To describe an 
octagon on a given straight line^ Fig. 45. 




A. 
Fig. 45 



B 
-Prob. XXXI. 



— Produce the given line ab both 
ways, and draw perpendiculars ae, 
BF; bisect the external angles a and 
B, by the lines ah, b c, which make 
equal to a b. Draw c D and H g par- 
allel to A E, and equal to a b ; from the 
centres g, d, with the radius a b, cut 
the perpendiculars at e, f, and draw 
E F to complete the octagon. 

Problem XXXIL — To convert a 
square into an octagon. Fig. 46. — Draw 
the diagonals of the square cutting at 



e-j from the comers a, b, c, d, with a e 
as radius, describe arcs cutting the 




Fig. 46.-Prob. XXXII. 

sides at g, h, &c.; and join the points 
so found to form the octagon. 

Problem XXXIIL — To inscribe 
an octagon in a circle, Fig. 47. — Draw 

B 




Fig. 47.-Prob. XXXIII. 

two diameters a c, b d at right angles ; 
bisect the arcs ab, bc, &c., at ^,/, 
&c., and join a ^, ^b, &c., to form the 
octagon. 

Problem XXXIV. — To describe 
an octagon about a circle, Fig. 48. — 




Fig. 48. -Prob. XXXIV. 

Describe a square about the given 
circle .A B, draw perpendiculars hk, 



12 



GEOMETRICAL PROBLEMS 



&c., to the diagonals, touching the 
circle, to form die octagon. 

Or, the points //, k, &c, may be 
found by cutting the sides from the 
comers of the square, as in the second 
last problem. 

Problem XXXV. — To describe a 
polygon of any number of sides upon a 
given straight Hne, Fig. 49. — Produce 




Fig. 49. — Prob. XXXV. 

the given line a b, and on a, with 
the radius a b, describe a semicircle, 
divide the semi-circumference into as 
many equal parts as there are to be 
sides in the polygon; say, in this ex- 
ample, five sides. Draw lines from 
A through the divisional points d, b, 
and r, omitting one point a, and on 
the centres b, d, with the radius a b, 
cut A ^ at E and a r at f. Draw 
D E, E F, F B to complete the polygon. 

Problem XXXVI. — To inscribe a 
circle within a polygon, Figs. 50, 51. — 
When the polygon has an even num- 
ber of sides, Fig. 50, bisect two op- 




Fig. sa— Prob. XXXVI. XXXVII. 

posite sides at a and b, draw a b, and 
bisect it at c by a diagonal d e; and 
with the radius ca describe the- circle. 



When the number of sides is odd, 
Fig. 51, bisect two of the sides at a 




Fig. 51.— Prob. XXXVI. XXXVII. 

and B, and draw lines a e, b d to the 
opposite angles, intersecting at c; 
from c, with the radius c a, describe 
the circle. 

Problem XXXVI I. — To describe a 
circle without a polygon. Figs. 50, 51. — 
Find the centre c as before, and \^dth 
the radius c d describe the circle. 

The foregoing selection of prob- 
lems on regular figures are the most 
usefiil in mechanical practice on that 
subject Several other regular figures 
may be constructed from them by 
bisection of the arcs of the circum- 
scribing circles. In this way a de- 
cagon, or ten-sided polygon, may be 
formed from the pentagon, as shown 
by the bisection of the arc b h at ^ 
in Fig. 41. Inversely, an equilateral 
triangle may be inscribed by joining 
the alternate points of division found 
for a hexagon. 

Problem XXXVIII.— 7?? inscribe 
a polygon of any number of sides 
within a circle, Fig. 52. — Draw the 
diameter a b, and through the centre 
E draw the perpendicular e c, cutting 
the circle at f. Divide e f into four 
equal parts, and set off three parts 
equal to those from f to c. Divide 
the diameter a b into as many equal 
parts as the polygon is to have sides; 
and from c draw c d through the 
second point of division, cutting the 
circle at d. Then a d is equal to one 



ON THE ELLIPSE. 



13 



side of the polygon, and by stepping 
round the circumference with the 




Fig. 52.— Proh. XXXVIII. 

length A D, the polygon may be com- 
pleted; 

The constructions for inscribing 
regular polygons in circles are suit- 
able also for dividing the circumfer- 
ence of a circle into a number of 
equal parts. To supply a means of 
dividing the circumference into any 
number of parts, including cases not 
provided for in the foregoing prob- 
lems, the annexed table of angles 
relating to polygons, expressed in 
dqgrees, will be found of general 
utib'ty. In tliis table the angle at 

Table of Polygonal Angles. 



Number 


Angle 


Number 


Angle 


, ofSidcs. 


at Centre. 


of Sides. 


at Centre. 


No. 


Degrees. 


No. 


Degrees. 


3 


120 


12 


30 


4 


90 


13 


27A 


5 


72 


H 


25f 


6 


60 


15 


24 


7 


5if 


16 


22^ 


8 


45 


17 


«^ 


9 


40 


18 


20 


10 

1 


36 


19 


19 


II 

1 


32A 


20 


18 



the centre is found by dividing 360°, 
the number of degrees in a circle, by 
the number of sides in the polygon ; 
and by setting off round the centre 
of the circle a succession of angles 
by means of the protractor, equal to 
the angle in the table due to a given 



number of sides, the radii so drawn 
will divide die circimiference into the 
same number of parts. The triangles 
thus formed are termed the elemen- 
tary triangles of the polygon. 

•Problem XXXIX. — To inscribe 
any regular polygon in a given circle; 
or to divide the circumference into a 
^ven number of equal parts, by means 
of ihe angle at the centre. Fig. 53. — 




Fig. 53.-Prob. XXXIX. 

Suppose the circle is to contain a 
hexagon, or is to be divided at the 
circumference into six equal parts. 
Find the angle at the centre for a 
hexagon, or 60°; draw any radius b c, 
and set off, by a protractor or other- 
wise, the angle at the centre cbd 
equal to 60°; then the interval cd is 
one side of the figure, 6x segment of 
the circumference; and the remaining 
points of division maybe found either 
by stepping along the circumference 
with the distance c d in the dividers, 
or by setting off the remaining five 
angles, of 60^ each, round the centre. 



PROBLEMS ON THE ELLIPSE. 

An ellipse is an oval figure, like a 
circle in perspective. The line a b, 
Fig. 54, that divides it equally in the 
direction of its greatest dimension, 
is the transverse axis; and the per- 
pendicular CD, through the centre, 
is the conjugate axis. Two points, 
F, G, in the transverse axis, are the 



14 



GEOMETRICAL PROBLEMS 



foci of the curve, each being called a 
focus; being so placed that the sum 
of their distances from either end of 
the conjugate axis, c or d, is equal 




F'g- 54* — Prob. XL. 

to the transverse axis. In general, the 
sum of their distances from any other 
point in the curve is equal to the 
transverse axis. A line drawn at 
right angles to either axis, and termi- 
nated by the curve, is a double ordi- 
nate^ and each half of it is an ordinate. 
The segments of an axis between an 
ordinate and its vertices are called 
abscisses. The double ordinate drawn 
through a focus is called the para- 
ineter of the axis. 

The squares of any two ordinates 
to the transverse axis, are to each 
other as the rectangles of their respec- 
tive abscisses. 

Problem XL. — To describe an el- 
lipse when the length and breadth are 
^veny Fig. 54. — On the centre c, with 
A £ as radius, cut the axis a b at f 
and G, the foci ; fix a couple of pins 
into the axis at f and G, and loop on 
a thread or cord upon them equal in 
length to the axis a b, so as when 
stretched to reach to the extremity c 
of the conjugate axis, as shown in dot- 
lining. Place a pencil or drawpoint 
inside the cord, as at h, and guiding 
the pencil in this way, keeping the 
cord equally in tension, carry the 
pencil round the pins f, g, and so 
describe the ellipse. 

Note, — This method is employed 



in setting off elliptical garden-plots, 
walks, &c. 

2d Method, Fig. 55. — Along the 
straight edge of a slip of stiff paper, 
mark off a distance a c equal to a c, 




Piff- 55.— Prob. XL. 2d method. 

I 

half the. transverse axis; and from 
the same point a distance a b equal 
to c D, half the conjugate axis. Place 
the slip so as to bring the point b on 
the line a b of the transverse axis, and 
the points on the line de; and set 
off on the dra\ving the position of the 
point a. Shifting the slip, so that 
the point b travels on the transverse 
axis, and the point c on the conjugate 
axis, any number of points in the 
curve may be found, through which 
the curve may be traced. 

id Metlwd^ Fig. 56. — ^The action 
of the preceding method may be em- 




Fig. 56.— -Prob. XL. 3d method. 

bodied so as to afford the means of 
describing a large curve continuously, 
by means of a bar mky with steel 
points w, /, k, rivetted into brass slides 
adjusted to the length of the semi- 
axes, and fixed with setscrews. A rec- 
tangular cross E G, with guiding slots, 
is placed coinciding with the two 



ON THE ELLIPSE. 



IS 



axes of the ellipse, ac and bh; by 
sliding the points ky /, in the slots, and 
carrying round the point m, the curve 
may be continuously described. A 
pen or pencil may be fixed at m, 

4M Method, Fig. 57. — Bisect the 
transverse axis at c, and through c 




Fig. 57. — Prob. XL. 4th method. 

draw the perpendicular d e, making 
CD and CE each equal to half the 
conjugate axis. From d or e, with 
the radius a c, cut the transverse axis 
at r, f', for the foci. Divide a c into 
a number of parts at the points i, 2, 
3, &c. With the radius a i, on f and 
f' as centres, describe arcs ; and with 
the radius b i, on the same centres, 
cut these arcs as shown. Repeat the 
operation for the other divisions of 
the transverse axis. The series of 
intersections thus made are points in 
the curve, through which the curve 
may be traced. 

5/>i Method, Fig. 58.— On the two 












^••■•' 


— •« 


«»«^ 
^•» 




^^ 






*-. 


, 


D 




*^ 


» _^^^^^^ 


^^ 




% 


» ^^ 






X 






• 




ly * * » 




1 
* 


\: \. 


id ' * * 






« • V 


if * ' * 




• 


II 


/• u 




1 


■% 


f * • 




! 


 1  


4i«; ? 


C_ 


1 
i 
 


' • 1 


*v» ' • •• • 




1 
1 
• 
» 
• 
1 
• 
1 


* * 1 
' * J 
1 * f 


A • ^k • • * t 




1 


» \ y f 


'-^Jv--y' J 




t 




• ^rs-* •» ! 




% 


'O*^ / 


\ : '^^^^IL.J' 




\AL 






T 


• • 




**——_. 


T 







of points, a, b, &c., in the circumfer- 
ence A F B, draw radii cutting the in- 
ner circle at d, b\ &c. From a, by 
&c., draw perpendiculars to ab; and 
from a'y ^', &c., draw parallels to a b, 
cutting the respective perpendiculars 
at «, Oy &c. The intersections are 
points in the curve, through which 
the curve may be traced. 

6/// Metlwdy Fig. 59. — When the 
transverse and conjugate diameters 




Fig. 58.— Prob. XL. 5th method. 

axes A B, D E as diameters, on centre 
c, describe circles; from a number 



Fig. 59.— Prob. XL. 6ih method. 

are given, a b, c d, draw the tangent 
E F parallel to a b. Produce c d, 
and on the centre c, with the radius 
of half a B, describe a semicircle hdk; 
from the centre g draw any number 
of straight lines to the points e, r, 
&c., in the line e f, cutting the cir- 
cumference at /, w,«, &c.; from the 
centre o of the ellipse draw straight 
lines to the points e, r, &c., and 
from the points /, w, «, &c., draw 
parallels to g c, cutting the lines o e, 
or, &c., at L, M, N, &C. These are 
points in the circumference of the 
ellipse, and the curve may be traced 
through them. Points in the other 
half of the ellipse are formed by ex- 
tending the intersecting lines as indi- 
cated in the figure. 

Problem XLL — To describe an 
ellipse approximately by means of cir- 
cular arcs. — First, with . arcs of two 
radii. Fig. 60. — Find the difference 



i6 



GEOMETRICAL PROBLEMS 



of the two axes, and set it off from the 
centre o to a and r, on oa and oc; 




Fig^. 60.— IVob. XLI. 

draw a c, and set off half ac to d; 
draw di parallel to ac^ set off o^ 
equal to o//, join ei, and draw the 
parallels e m, d m. From m, with 
radius m c, describe an arc through 
c; and from / describe an arc through 
D ; from d and e describe arcs through 
A and B. The four arcs form the 
ellipse approximately. 

Note. — ^This method does not ap- 
ply satisfactorily when the conjugate 
axis is less than two-thirds of the 
transverse axis. 



o M equal to c l, and on d describe 
an arc with radius dm; on a, with 
radius o l, cut this arc at a. Thus 
the five centres d, a, by h, h' are found, 
from which the arcs are described to 
form the ellipse. 

Note, — ^This process works well for 
nearly all proportions of ellipses. It 
is employed in striking out vaults and 
stone bridges. 

Problem XLII. — To draw a tan- 





'K o'. 



%^» / 









\ 



V. 



Fig. 6x. — Prob. XLI. ad method. 

Second, with arcs of three radii, 
Fig. 61. — On the transverse axis ab 
draw the rectangle b g, on the height 
o c ; to the diagonal a c draw the per- 
pendicular ghd; set off ok equal 
to c, and describe a semicircle on 
AK, and produce oc to l; set off 



Fig. 62. -Prob. XLII. 

gefit to an ellipse through a given point 
in theatrve. Fig. 62. — From the given 
point T draw straight lines to the 
foci F, f'; produce f t beyond the 
curve to c, and bisect the exterior 
angle ^ t f, by the line t //, which 
is the tangent. 

Problem XLIII. — To draw a 
tangent to an ellipse from a given 
point without the curve. Fig. 63. — 
From the given point t, with a 
radius to the nearest focus f, de- 
scribe an arc on the other focus 
' f', with a radius equal to the trans- 
verse axis, cut the arc at k l, and 




Fig. 63.-Prob. XLIII. 

draw K f', L f', cutting the curve at 
M, N. The lines t m, t n are tangents. 



ON THE PARABOLA. 



17 



PROBLEMS ON THE PARABOLA. 

A parabola, dac, Fig. 64, is a 
cun'e such that every point in the 
curve is equally distant from the di- 
rectrix K L and the focus r. The 
focus lies in the axis a b drawn from 
the v€rtex or head of the curve a, so 
as to divide the figure into two equal 
parts. The vertex a is equidistant 
from the directrix and the focus, or 
A^=AF. Any line parallel to the 
axis is a diameter. A straight line, 
as EG or D c, drawn across the figure 
at right angles to the axis is a double 
ordinate, and either half of it is ap 
ordinate. The ordinate to the axis 
E F G, drawn through the focus, is 
called iki't parameter of the axis. A 
s^ment of the axis, reckoned from the 
vertex, is an absciss of the axis; and 
it is an absciss of the ordinate drawn 
from the base of the absciss. Thus, 
A B is an absciss of the ordinate b c. 
Abscisses of a parabola are as the 
squares of their ordinates. 

Problem XLIV. — To describe a 
parabola when an absciss and its ordi- 
nate are given; that is to say, when 
the height and breadth are given, 
Fig. 64. — Bisect the given ordinate 





< 


^ 


J, 

a. 


J 


^ 




\, 


/• 


s 


\ 


\ 


/ 




\ 


V 


/ 


3 




$ \ 


:» 


J 
1 




<» c 



rig. 64.— Prob. XLIV. 

fic at tf ; draw A a, and then a b per- 
pendicular to it, meeting the axis at 
A Set off A if, A F, each equal to b b; 
and draw k ^ l perpendicular to the 
aiis. Then k l is the directrix and 
F is (he focus. Through f and any 



number of points, Oy Oy &c., in the 
axis, draw double ordinates, n «, 
&c. ; and on the centre f, with tlie 
radii ¥ e,oe, &c., cut the respective 
ordinates at e, g, «, «, &c. The curve 
may be traced through these points 
as shown. 

2d Method; by means of a square 
and a cord. Fig. 65. — Place a straight- 




Fig. 65.— Prob. XLIV. ad method. 

edge to the directrix e n, and apply 
to it a square leg. Fasten to the 
end G, one end of a thread or cord 
equal in length to the edge £ g, and 
attach the other end to the focus f; 
slide the square along the straight- 
edge, holding the cord taut against 
the edge of the square by a draw- 
point or pencil d, by which the curve 
is described. 

^d Method; when the height and the 
base are given, Fig. 66. — Let a b be 



J5 


p 








A 










F 


a 






J 


r 


tr 




^ 


V 






Of 


^ 




y 


r 










1 


V 




I 


c 


A 


f 














^ 


\ 





^ 


1. 






m 










^ 


\ 


a 


t 


^ i 


f t 


V 1 


^ C 


PI 


r < 


1 1 


\ 1 


? < 


i 1 


> 



Fig. 66.— Prob. XLIV. 3d method. 

the given axis, and c d a double ordi- 
nate or base; to describe a parabola 

2 



I8 



GEOMETRICAL PROBLEMS 



of which the vertex passes through a. 
Through a draw e f parallel to c d, 
and through c and d draw c e and 
D F parallel to the axis. Divide b c 
and BD into any number of equal 
parts, say five, at «, by &c., and divide 
c £ and D F into the same number of 
parts. Through the points a^ b, c, d 
in the base c d, on each side of the 
axis, draw perpendiculars, and through 
a, by Cy //, in c £ and d f, draw lines to 
the vertex a, cutting the perpendicu- 
lars at eyfygy h. These are points in 
the parabola, and the curve cad may 
be traced as shown, passing through 
them. 



PROBLEMS ON THE HYPERBOLA. 

The vertices a, b. Fig. 67, of oppo- 
site hyperbolas, are the heads of the 
curves, and are points in their centre 
or axial lines. The transverse axis 
A B is the distance between the ver- 
tices, of which the centre c is the 
centre. The conjugate axis g h is a 
straight line drawn through the centre 
at right angles to the transverse axis. 
An ordinate f k is a straight line 
drawn from any point of the curve 
perpendicular to the axis. The seg- 
ments of the transverse axis a f, b f, 
between an ordinate f k and the ver- 
tices of the curves, are abscisses. 
The parameter is the double ordinate 
drawn through the focus. The as- 
symptotes are two straight lines, s s, 
R R, drawn from the centre through 
the ends of a tangent ed at the vertex, 
equal and parallel to the conjugate 
axis, and bisected by the transverse 
axis. 

The nature of the hyperbola is such 
that the difference of the distances 
of any point in the curve from the 
foci is always the same, and is equal 
to the transverse axis. 

In a hyperbola the squares of any 
two ordinates to the transverse axes 
are to each other as the rectangles of 
their abscisses. 



Problem XLV. — To describe a 
hyperbolay the transverse and conjugate 
axes being givetiy Fig. 67. — Draw ab 




Fig. 67.— Prob. XLV. 

equal to the transverse axis, and d e 
perpendicular to it and equal to the 
conjugate G h. On c, with the radius 
c E, describe a circle cutting a b pro- 
duced, at f/; these points are the foci. 
In A B produced take any number of 
points Oy Oy &c., with the radii a^^ b^, 
and on centres f,/ describe arcs cut- 
ting each other at «, ;z, &c. These 
are points in the curve, through which 
it may be traced. 

2d Metliody Fig. 67. — The curve 
may be drawn thus: — Let the ends 
of two threads/p q, f p q, be fastened 
at the points /, f, and be made to 
pass through a small bead or pin p, 
and knotted together at q. Take 
hold of Q, and draw the threads tight ; 
move the bead along the threads, and 
the point ? will describe the curve. 
If the end of the long thread be fixed 
at F, and the short thread at /, the 
opposite curve may be described in 
the same ftianner. 

Or, the line /q may be replaced 
by a straight-edge turning on a pin 
at/ and the cord F q joined to it at Q. 
The curve may then be described by 
means of a point or pencil in the same 
manner as for the parabola. Fig. 65. 

3^/ Method; when the breadth c d. 



ON THE HYPERBOLA, CYCLOID, EPICYCLOID. 



19 



hd^ A B, and transverse axis a a! of 
the curve are gtvetiy Fig. 68. — Divide 




Fig. 68.— Protx XLV. 3d method. 

the base or double ordinate c d into 
a number of equal parts on each side 
of the axis at a, d, &c ; and divide 
the parallels c e, d f, into the same 
number of equal parts at a, d, &c. 
From the points a, d, &c., in the base, 
draw lines to a', and from the points 
a, b, &c., in the verticals, draw lines 
to A, cutting the respective lines from 
the base. Trace the curve through 
the intersections thus obtained. 



THE CYCLOID AND EPICYCLOID. 

Problem XLVL — To describe a 
cydoid^ Fig. 69. — When a wheel or a 
circle D G c rolls along a straight line 




Fig. 69.— Prob. XLVI. 



one revolution, it measures off a 
straight line a b exactly equal to the 
circumference of the circle d g c, 
which is called the generating circle, 
and a point or pencil fixed at the 
point D in the circumference traces 
out a curvilinear path a d b, called a 
cycloid, A B is the bcLse and c D is the 
axis of the cycloid. 

Place the generating circle in the 
middle of the cycloid, as in the figure, 
draw a line e h parallel to the base, 
cutting the circle at g; and the tan- 
gent H I to the curve at the point h. 
Then the following are some of the 
properties of the cycloid : — 

The horizontal line h G=arc of the 
circle G d. 

The half-base a c =» the half-circum- 
ference c G D. 

The arc of the cycloid d h = twice 
the chord d g. 

The half-arc of the cycloid d a = 
twice the diameter of the circle d c. 

Or, the whole arc of the cycloid 
A D B = four times the axis c d. 

The area of the cycloid a d b a = 
three times the area of the generating 
circle d c 

The tangent H i is parallel to the 
chord G D. 

Problem XLVII. — To describe an 




Fig. 7o.-Prob. XLVIL 

A B, Fig. 69, beginning at a and end- exterior epicycloid ^ Fig. 70. — The epicy- 
ing at D, where it has just completed chid differs from the cycloid in this, 



20 



GEOMETRICAL PROBLEMS 



that it is generated by a point d in 
one circle do rolling upon the cir- 
cumference of another circle a c b, 
instead of on a flat surface or line; 
the former being the generating circle^ 
and the latter the fundametital circle. 
The generating circle is shown in four 
positions, in which the generating 
point is successively marked d, d', d", 
d'^'. a d"' b is the epicycloid. 

Problem XLVIIL — To describe 




Fig. 71.— Prob. XLVIIL 

an interior epicycloid^ Fig. 71. — If the 
generating circle be rolled on the in- 
side of the fundamental circle, as in 
Fig. 71, it forms an interior epicycloid^ 
or hypocycloidy a j>"' b, which becomes 
in this case nearly a straight line. The 
other points of reference in the figure 
correspond to those in Fig. 70. 
When the diameter of the generating 
circle is equal to half that of the fun- 
damental circle, the epicycloid be- 
comes a straight line, being in fact a 
diameter of the larger circle. 



THE CATENARY. 

« 

When a perfecdy flexible string, or 
a chain consisting of short links, is 
suspended from two points m, n, Fig. 
72, it is stretched by its own weight, 
and it forms a curve line known as 
the catenary, m c n. The point c, 
where the catenary is horizontal, is 
the vertex. 

Problem XLIX. — To describe a 



catenary^ Fig. 72. — Draw the vertical 
c G equal to the length of the arc of the 
chain, m c, on one side of the vertex, 
and divide it into a great number of 
equal parts,at ( i ), ( 2), (3 ),&c. Draw the 
horizontal line c h equal to the length 
of so much of the rope or chain as 
measures by its weight the horizontal 
tension of the chain. From the point 
c as the vertex, set off" c (i) on the 
horizontal line equal to c i on the 
vertical; and (i) (2) from the point 
(i), parallel to h i and equal to c(i); 
and again (2) (3) from the point (2) 
parallel to h 2 and equal to c (i); 
and so on till the last segment (6) m 
is drawn parallel to h g. The poly- 
gon c (i) (2) (3) . . . M, thus formed, 
is approximately the catenary curve, 
which may be traced through the 
middle points of the sides of the 
polygon. A similar process being 
performed for the other side of the 
curve, the catenary is completed. 




Fig. 7a.— Prob. XLIX. 

2d Method. — Suspend a finely 
linked chain against a vertical wall. 
The curve may be traced from it, on 
the wall, ^swering the conditions of 
given length and height, or of given 
width or length of arc, A cord having 
numerous equal weights suspended 
from it at short and equal distances 
may be used. 



CIRCLES, PLANE TRIGONOMETRY. 



21 



CIRCLES. 

The circumference of a circle is commonly signified in mathematical 
discussions by the symbol x, which indicates the length of the circumfer- 
ence when the diameter t= i. 

The area of a circle is as the square of the diameter, or the square of the 
circumference. 

The ratio of the diameter to the circumference is as i to 3*141593 — 

commonly abbreviated, as i to 3'i4i6 

approximately, as i to 3I 

or as 7 to 22 

WTien the diameter = i, the area is equal to 785398 + 

or, commonly abbreviated, 7854 

approximately, j^ths. 

HTien the circumference = i, the area is equal to "079577 + 

or, abbreviated, , '0796 

approximately, A^hs, or '08. 

In these ratios, the diameter and the circumference are taken lineally, 
and the area superficially. So that if the diameter = i foot, the circum- 
ference is equal to 3'i4i6 feet, and the area is equal to 7854 square foot 

Note, — If the first three odd figures, 1,3, 5, be each put down twice, the 
first three of these will be to the last three, that is 113 is to 355, as the 
diameter to the circumference. 



PLANE TRIGONOMETRY. 



The circumference of a circle is supposed to be divided into 360 degrees 
or divisions, and as the total angularity about the centre is equal to four right 
angles, each right angle contains 90 degrees, or 90°, and half a right angle 




<- 

Fig. 73. — Definidons in Plane Trigonometry. 






contains 45**. Each degree is divided into 60 minutes, or 6o'j and, for the 
sake of still further minuteness of measurement, each minute is divided into 
60 seconds, or 60", In a whole circle there are, therefore, 360 x 60 x 60 = 



22 GEOMETRICAL PROBLEMS. 

1,296,000 seconds. The annexed diagram, Fig. 73, exemplifies the rela- 
tive positions of the sine, cosine, versed sine, tangent, co-tangent, secant, 
and co-secant of an angle. It may be stated, generally, that the correlated 
quantities, namely, the cosine, co-tangent, and co-secant of an angle, are 
the sine, tangent, and secant, respectively, of the complement of the given 
angle, the complement being the difference between the given angle and a 
right angle. The supplement of an angle is the amount by which it is 
less than two right angles. 

When the sines and cosines of angles have been calculated (by means of 
formulas which it is not necessary here to particularize), the tangents, co-tan- 
gents, secants, and co-secants are deduced from them according to the 
following relations : — 

rad. X sin. rad/^ rad.*^ rad.^ 

tan. = ; cotan. = ; sec. = ; cosec. = . 

COS. tan. COS. sin. 

For these the values will be amplified in tabular form. 

A triangle consists of three sides and three angles. When any three 
of these are given, including a side, the other three may be found by cal- 
culation : — , 

Case i. — IVAm a side and its opposite angle are two of the ^vm parts. 

Rule i. To find a side, work the following proportion: — 
as the sine of the angle opposite the given side 
is to the sine of the angle opposite the required side, 
so is the given side 
to the required side. 

Rule 2. To find an angle: — 

as the side opposite to the given angle 
is to the side opposite to the required angle, 
so is the sine of the given angle 
to the sine of the required angle. 

Rule 3. In a right-angled triangle, when the angles and ofie side tiext t/ie 
right angle are given, to find the other side: — 
as radius 

is to the tangent of the angle adjacent to the given side, 
so is this side 
to the other side. 

Case 2. — When two sides and the included angle are given. 

Rule 4. To find the other side: — 

as the sum of the two given sides 

is to their difference, 

so is the tangent of half the sum of their opposite angles 

to the tangent of half their difference — 
add this half difference to the half sum, to find the greater angle; and 
subtract the half difference from the half sum, to find the less angle. Tlie 
other side may then be found by Rule i. 

Rule 5. When the sides of a right-angled triangle are givai, to find the 
angles: — , 



MENSURATION OF SURFACES. 23 

as one side 

is to the other side, 

so is the radius 

to the tangent of the angle adjacent to the first side. 

Case 3. — When the three sides are given. 

Rule 6. To find an angle. Subtract the sum of the logarithms of the 
sides which contain the required angle, from 20; to the remainder add the 
logarithm of half the sum of the three sides, and that of the difference 
bet^s'ccn this half sum and the side opposite to the required angle. Half 
the sum of these three logarithms will be the logarithmic cosine of half the 
required angle. The other angles may be found by Rule i. 

Rule 7. Subtract the sum of the logarithms of the two sides which con- 
tain the required angle, from 20, and to the remainder add the logarithms 
of the differences between these two sides and half the sum of the three 
sides. Half the result will be the logarithmic sine of half the required 
angle. 

Note, — In all ordinary cases either of these rules gives sufficiently accur- 
ate results. It is recommended that Rule 6 should be used when the 
required angle exceeds 90°; and Rule 7 when it is less than 90°. 



MENSURATION OF SURFACES. 

To find the area of a paraiieiogram. Multiply the length by the height, or 
perpendicular breadth. 

Or, multiply the product of two contiguous sides by the natural sine 
of the included angle. 

To find the area of a triangle. Multiply the base by the perpendicular 
height, and take half the product 

Or, multiply half the product of two contiguous sides by the natural 
sine of the included angle. 

To find the area of a trapezoid. Multiply half the sum of the parallel 
sides by the perpendicular distance between them. 

To find the area of a quadrilateral inscribed in a circle. From half the 
sum of the four sides subtract each side severally; multiply the four re- 
mainders together; the square root of the product is the area. 

To find the area of any qiiadrilateral figure. Divide the quadrilateral into 
two triangles; the sum of the areas of the triangles is the area. 

Or, midriply half the product of the two diagonals by the natural sine of 
the angle at their intersection. 

Note, — As the diagonals of a square and a rhombus intersect at right angles 
(the natural sine of which is i), half the product of their diagonals is the 
area. 

To find the area of any polygon. Divide the polygon into triangles and 
trapezoids by drawing diagonals; find the areas of these as above shown, 
for the area. 

To find the area of a regular polygon. Multiply half the perimeter of the 
polygon by the perpendicular drawn from the centre to one of the sides. 
Nc^e, — ^To find the perpendicular when the side is given — 



24 



GEOMETRICAL PROBLEMS. 



as radius 

to tangent of half-angle at perimeter (see table No. i), 
SO is half length of side 
to perpendicular. 
Or, multiply the square of a side of any regular polygon by the corres- 
ponding area^in the following table: — 

Table No. i. — ^Angles and Areas of Regular Polygons, 



Name. 


Number 

of 
Sides. 


One half 

Angle at the 

Perimeter. 


Area. 
(Side=i) 

1 


Perpendi- 
cular. 
(Side = I) 


Equilateral triangle, 

Square, 


3 
4 

5 
6 

7 
8 

9 

lO 

II 

12 


30° 
45° 

54° 
60° 

64°l 
67°i 
70° 
72° 

73°A 
75° 


0-4330 
I -oooo 

17205 

2-5981 

3*6339 
4-8284 

6I8I8 

76942 

93656 

II-I962 


0-2887 
0-5000 
0-6882 
0-8660 
1*0383 
1-2071 

1*3737 

1-5388 
17028 

I -8660 


.uri|^i.M>&^, .•■.«•. 

Pentagon 


Hexagon 


Heptagon 


Octagon 


Nonagon 


Decagon 


Undecagon 


Dodecagon 







To find the circumference of a circle. Multiply the diameter by 3 -14 16. 
Or, multiply the area by 12*5664; the square root of the product is the 
circumference. 

To find the diameter of a circle. Divide the circumference by 3-1416. 
Or, multiply the circumference by '3183. 

Or, divide the area by 7854; the square root of the quotient is the 
diameter. 

To find the area of a circle. Multiply the square of the diameter by 7854. 
Or, multiply the circumference by one-fourth of the diameter. 
Or, multiply the square of the circumference by '07958. 

To find the length of an arc of a circle. Multiply the number of degrees 
in the arc by the radius, and by '01745. 

Or, the length may be found nearly, by subtracting the chord of the whole 
arc from eight times the chord of half the arc, and taking one-third of the 
remainder. 

To find the area of a sector of a circle. Multiply half the length of the 
arc of the sector by the radius. 

Or, multiply the number of degrees in the arc by the square of the radius, 
and by -008727. 

To find the area of a segment of a circle. Find the area of the sector 
which has the same arc as the segment; also the area of the triangle 
formed by the radial sides of the sector and the chord of the arc; the 
difference or the sum of these areas will be the area of the segment, ac- 
cording as it is less or greater than a semicircle. 

To find the area of a ring included between the circumferences of two con- 



MENSURATION OF SURFACES. 



25 



oniric circles. Multiply the sum of the diameters by their difference, and 
577854. • 

To find Ike area of a cycloid. Multiply the area of the generating circle 

To find tJu length of an arc of a parabola^ cut off by a double ordinate 
to the axis. To the square of the ordinate add four-fifths of the square of 
the absciss; twice the square root of the sum is the length nearly. . 

Note. — ^This rule is an approximation which applies to those cases only 
in which the absciss does not exceed half the ordinate. 

To find the area of a parabola. Multiply the base by the height; two- 
thirds of the product is the area. 

To find the circumference of an ellipse. Multiply the square root of half the 
sum of the squares of the two axes by 3*1416. 

To find the area of an ellipse. Multiply the product of the two axes by 

7^54. 

Note, — ^The area of an ellipse is equal to the area of a circle of which 
the diameter is a mean proportional between the two axes. 

To find the area of an elliptic segment, the base of which is parallel to 
either axis of the ellipse. Divide the height of the segment by the axis of 
which it is a part, and find the area of a circular segment, by table No. 
VII., of which the height is equal to this quotient; multiply the area thus 
found by the two axes of the ellipse successively; the product is the area. 

To find the length of cm arc of a hyperbola^ beginning at the vertex. To 
19 times the transverse axis add 21 times the parameter to this axis, and 
multiply the sum by the quotient of the absciss divided by the transverse. 
2(L To 9 times the transverse add 2 1 times the parameter, and multiply the 
sum by the quotient of the absciss divided by the transverse. 3d. To 
each of these products add 15 times the parameter, and then 

as the latter sum 

is to the former sum, 

so is the ordinate 

to the length of the arc, nearly. 

To find the area of a hyperbola. To the product of the transverse and 
absciss add five-sevenths of the square of the absciss, and multiply the square 
root of the sum by 21; to this product add 4 times the square root of 
the product of the transverse and absciss; multiply the sum by 4 times the 
product of the conjugate and absciss, and divide by 75 times the transverse. 
The quotient is the area nearly. 

To find the area of any cundlinecU figure, 
hounded at the ends by parallel straight lines, 
Fig. 74. Divide the length of the figure 
ab into any even number of equal parts, 
and draw oidinates c, d, e, &c, through the 
pomts of division, to touch the boundary 
lines. Add together the first and last 
ordinates {c and k), and call the sum a; 
add together the even ordinates (that is, 
^Jy ^y ), and call the sum b; add together 

the odd ordinates, except the first and last (e,g, i), and call the sum c Let 
D be the common distance of the ordinates, then 





Fig. 74.— For Area of Curvilinear Figure. 



26 



GEOMETRICAL PROBLEMS. 



(a + 4 B + 2 c) 



X D = area of figure. 



This IS known as Simpson's Rule. 

2d Method^ Fig. 74. — Having divided the figure into an even or an odd 
number of equal parts, add together the first and last ordinates, making 
the sum a; and add together all the intermediate ordinates, making the 
sum B. Let l = the length of the figure, and n = the number of divisions, 
then 



A + 2B 

2n 



X L = area of figure. 



That is to say, twice the sum of the intermediate ordinates, plus the first 
and last ordinates, divided by twice the number of divisions, and multi- 
plied by the length, is equal to the area of the figure. 

This method is that commonly used; it is sufficiently near to exactness 
for most purposes. 

3^ Methody Fig. 74. — Having divided the' figure as above, measure by a 
scale the mean depth of each division, at the middle of the division; add 
together the depths of all the divisions, and divide the sum by the number 
of divisions, for the average depth; multiply the average depth by the length, 
which gives the area. 

For the sake of obtaining a more nearly exact result, the figure may be 
divided into two half-parts, c^k^ Fig. 75, one at each end, and a number 
of whole equal parts, d^e^f^gji^ij^ intermediately. Then the ordinates 
separating these parts, excluding the extreme ordinates, may be measured 



» 





Fig. 75. 



For- Area of Cunrilincal Figures. 



Fig. 76. 



direct, and the sum of the measurements divided by the nimiber of them, 
and multiplied by the length, for the area. 

Note, — In dealing with figures of excessively irregular outline, as in Fig. 
76, representing an indicator-diagram from a steam-engine, mean Hnes, ab^ 
c dy may be substituted for the actual lines, being so traced as to intersect 
the undulations, so that the total area of the spaces cut off may be com- 
pensated by that of the extra spaces inclosed. 

Note 2. — The figures have been supposed to be bounded at the ends by 
parallel planes. But they may be terminated by curves or angles, as in 
Fig. 76, at ^, when the extreme ordinates become nothing. 



MENSURATION OF SOLIDS. 2^ 



MENSURATION OF SOLIDS. 

To find tke surface of a prism or a cylinder. The perimeter of the end 
multiplied by the height gives the upright surface ; add twice the area of 
an end. 

Tofijid the cubic contents of a prism or a cylinder. Multiply the area of 
the base by the height 

To find the surface of a pyramid or a cone. Multiply the perimeter of the 
base by half the slant height, and add the area of the base. 

To find the aibic contents of a pyramid or a cone. Multiply the area of 
the base by one-third of the perpendicular height. 

To find the surface of afrustrum of a pyramid or a cone. Multiply the 
sum of the perimeters of the ends by half the slant height, and add the 
areas of the ends. 

To find the cubic cofitents of a frustrum of a pyramid, or a cone, — Add 
together the areas of the two ends, and the mean proportional between them 
(that is, the square root of their product), and multiply the sum by one-third 
of the perp>endicular height. 

Or, when the ends are circles, add together the square of each diameter, 
and the product of the diameters, and multiply the sum by 7854, and by 
one-third of the height 

To find the aibic contaits of a wedge, — To twice the length of the base 
add the length of the edge ; multiply the sum by the breadth of the base, 
and by one-sixth of the height 

To find the cubic contents of aprismoid {a solid of which the tivo etuis are dis- 
similar but parallel plane figures of the same number of sides), — To the sum 
of the areas of the two ends, add four times the area of a section parallel to 
and equally distant from both ends; and multiply the sum by one-sixth of 
the length 

Note. — ^This rule gives the true content of all fmstrums, and of all solids 
of which the parallel sections are similar figures; and is a good approxima- 
tion for other kinds of areas and solidities. 

To find the surface of a sphere. — Multiply the square of the diameter by 
31416. 

Note. — ^The surface of a sphere is equal to 4 times the area of one of its 
great circles. 

2. The surface of a sphere is equal to the convex surface of its circum- 
scribing cylinder. 

3. The surfaces of spheres are to one another as the squares of their 
diameters. 

To find the curve surface of any segmmt or zone of a sphere. — Multiply the 
diameter of the sphere by the height of the zone or segment, and by 3*1416. 

Note. — The curve surfaces of segments or zones of the same sphere are 
to one another as their heights. 

To find the cubic contents of a sphere, — Multiply the cube of the diameter 
by -5236. 
Or, multiply the surface by one-sixth of the diameter. 



28 GEOMETRICAL PROBLEMS. 

Note. — The contents of a sphere are two-thirds of the contents of its 
circumscribing cylinder. 

2. The contents of spheres are to one another as the cubes of their 
diameters. 

To find the aibic contents of a segment of a sphere. — From 3 times the 
diameter of the sphere subtract twice the height of the segment; multiply 
the difference by the square of the height, and by '5236. 

Or, to 3 times the square of the radius of the base of the segment, add 
the square of its height; and multiply the sum by the height, and by '5236. 

To find the cubic contents of a frustrum or zone of a sphere, — ^To the sum 
of the squares of the radii of the ends add Yi of the square of the height; 
multiply the sum by the height, and by 1*5708. 

To find the cubic contents of a spheroid. — Multiply the square of the re- 
volving axis by the fixed axis and by '5236. 

Note. — ^The contents of a spheroid are two-thirds of the contents of its 
circumscribing cylinder. 

2. If the fixed and revolving axes of an oblate spheroid be equal to the 
revolving and fixed axes of an oblong spheroid respectively, the contents of 
the oblate are to those of the oblong spheroid as the greater to the less axis. 

To find the cubic contents of a segment of a spheroid. — ist. When the base 
is parallel to the revolving axis. Multiply the difference between thrice the 
fixed axis and double the height of the segment, by the square of the height, 
and the product by '5236. Then, 

as the square of the fixed axis 

is to the square of the revolving axis, 

so is the last product 

to the content of the segment 
2d. When the base is perpendicular to the revolving axis. Multiply the 
diflference between thrice the revolving axis and double the height of the 
segment, by the square of the height, and the product by '5236. Then, 

as the revolving axis 

is to the fixed axis, 

so is the last product 

to the content of the segment. 

To find tlie solidity of the middle frustrum of a spheroid. — ist When the 
ends are circular, or parallel to the revolving axis. To twice the square of 
the middle diameter, add the square of the diameter of one end; multiply 
the sum by the length of the frustrum, and the product by '2618 for the 
content. 

2d. When the ends are elliptical, or perpendicular to the revolving axis. 
To twice the product of the transverse and conjugate diameters of the middle 
section, add the product of the transverse and conjugate diameters of one 
end; multiply the sum by the length of the frustrum, and by '2618 for the 
content. 

To find tJie cubic contents of a parabolic conoid. — Multiply the area of the 
base by half the height. 

Or, multiply the square of the diameter of the base by the height, and 
by -3927. 

To find the cubic contents of a frustrum of a parabolic cofioid. — Multiply 
half the sum of the areas of the two ends by the height of the frustrum. 



MENSURATION OF SOLIDS. 



29 



Or, muUiply the sum of the squares of the diameters of the two ends by 
the height, and by '3927. 

Tofifid the cubic contents of a parabolic spindle, — Multiply the square of the 
middle diameter by the length, and by -41888. 

To find the cubic contents of the middle frustrum of a parabolic spindle, — 
Add together 8 times the square of the largest diameter, 3 times the square 
of the diameter at the ends, and 4 times the product of the diameters; 
multiply the sum by the length of the frustrum, and by '05236. 

To find the surface and the cubic contmts of any of the five regular solids^ Figs. 




Fig- 77- 



Fig. 78. 



Fig. 79- 



Fig. 80. 



Fig. 81. 



77, 78, 79, 80, 81. — For the surface, multiply the tabular area below, by the 
square of the edge of the solid. 

For the contents, multiply the tabular contents below, by the cube of the 
given edge. 

Note. — ^A regular solid is bounded by similar and regular plane figures. 
There are five regular solids, shown by Figs. 77 to 81, namely: — 

The tetrahedron^ bounded by four equilateral triangles. 

The hexahedron, or cube, bounded by six squares. 

The octahedron^ bounded by eight equilateral triangles. 

The dadecahedron,\>o\ixidtd by twelve pentagons. 

The icosahedron^ bounded by twenty equilateral triangles. 

Regular solids may be circumscribed by spheres; and spheres may be 
inscribed in regular solids. 

Surfaces and Cubic Contents of Regular Solids. 



Number 
of sides. 


Name. 


Area. 
Edge = I. 


Contents. 
Edge=i. 


4 
6 

8 

12 
20 


Tetrahedron 

Hexahedron 

Octahedron '. . 

Dodecahedron 

Icosahedron 


1:7320 
6*0000 

3*4641 
20*6458 

86603 


0*1178 
I *oooo 
0*4714 
7*6631 
2*1817 



To find the cubic contents of an irregular solid, — Suppose it divided into 
parts, resembling prisms or other bodies measurable by preceding rules; find 
the content of each part; the sum of the contents is the cubic contents of 
the solid. 

Note, — The content of a small part is found nearly by multiplying half 
the sum of the areas of each end by the perpendicular distance between 
them. 



30 GEOMETRICAL PROBLEMS. 

Or, the contents of small irregular solids may sometimes be found by im- 
mersing them under water in a prismatic or cylindrical vessel, and observing 
the amount by which the level of the water descends when the solid is 
withdrawn. The sectional area of the vessel being multiplied by the descent 
of the level, gives the cubic contents. 

Or, when the solid is very large, and a great degree of accuracy is not 
requisite, measure its length, breadth, and depth in several different places, 
and take the mean of the measurement for each dimension, and multiply 
the three means together. 

Or, when the surface of the solid is very extensive, it is better to divide 
it into triangles, to find the area of each triangle,' and to multiply it by the 
mean depth of the triangle for the contents of each triangular portion ; the 
contents of the triangular sections are to be added together. 

The mean depth of a triangular section is obtained by measuring the 
depth at each angle, adding . together the three measurements, and taking 
one-third of the sum. 



MENSURATION OF HEIGHTS AND DISTANCES. 

To find the height of an accessible object. — Measure the distance from the 
base of the object to any convenient station on the same horizontal plane; 
and at this station take the angle of altitude. Then 

as radius 

to tangent of the angle of altitude, 

so is the horizontal distance 

to the height of the object above the horizontal plane passing 
through the eye of the observer. Add the height of the eye, and the sum 
is the height of the object. 

Note, — The station should be chosen so that the angle of altitude should 
be as near to 45° as practicable; because the nearer to 45'', the less is the 
error in altitude arising from error of observation. 

When the angle of elevation is 45^ the height above the plane of the 
eye is equal to the distance. When it is 26° 34', the height is half the dis- 
tance. 

To find approximately the height of an accessible object. — There are four 

methods based on the principle of similar triangles. 
I St. By a geometrical square^ Fig. 82. — This is a 
square, a b, with two sights on one of its sides, a ;/, 
a plumb-line hung from one extremity, ;/, of that 
side, and each of the twp sides opposite to that 
extremity, mb,ma^ divided into 100 equal parts; 
the division beginning at the remote ends, so that 
the 1 00th divisions meet at the corner m. Let re 
be the object, and the sights be directed to the 
summit ^, at the known distance ad. When the 
Fig. 82.-Mcnsuration of a plummet cuts the side b m at, say, c, then by similar 

triangles, nb\nc\\ad\de. Or, if the plumb-line 
cuts the side a m, then the part of a m cut off is to <z « : : ad\ de. Adding 
to de the height of the eye rd, the sum is the height of the object, re. 




MENSURATION OF HEIGHTS AND DISTANCES. 



31 



2d. By shadows. Fig. 83.~7When the sun shines, fix a pole ^^ in the 
ground, vertically, and measure its shadow a b. Measure also the shadow de 




Fig. 83. 



Mensuration of a Height. 




of the object € m; then, by similar triangles, ab',bc\:de\ e m^ the height of 
the object. 

3d. By r^4ctiofiy Fig. 84. — Place a basin of water, or any horizontal 
reflecting surface, at a, level with the base of the object de, and retire from 
it till the eye at c sees the top of the object e, 
in the centre of the basin at a. Then, by similar 
triangles, abi bc\\ad\de, 

4tt By two poles. Fig. 85. — Fix two poles a m, 
cfiy of unequal lengths, parallel to the object er, 
so that the eye of the observer at a, the top of 
the shorter f>ole, may see c, the top of the longer 
pole, in a line with e, the summit of the object re. 
By similar triangles, ab \bc\\ad\de\ and adding 
rd, the height of the eye, to de, the sum r^ is the 
height of the object. 

To find the distance of the visible horizon. — To 
half the logarithm of the height of the eye, add 3*8105; the sum is the 
logarithm of the distance in feet, nearly. 

To find the distance of an object by the motion of sound, — Multiply the 
number of seconds that elapse between the flash or other sign of the gene- 
ration of the sound and the arrival of the sound to the ear, by 1120. The 
product is the distance in feet. 

Note. — ^\Vhen a sound generated near the ear returns as an echo, half the 
interval of time is to be taken, to find the distance of the reflecting surface. 




Fig. 85. Menmration of a 
Height. 



32 MATHEMATICAL TABLES. 



MATHEMATICAL TABLES. 

Table No. I. — Of Logarithms of Numbers from i to 10,000. 

Logarithms consist of integers and decimals; but, for the sake of com- 
pactness, the integers have been omitted in the table, except in the short 
preliminary section containing the complete logarithms of numbers from i to 
100. The table No. I. contains the decimal parts, to six places, of the loga- 
rithms of numbers from i to 10,000. The integer, or index, or character- 
istic of a logarithm, standing on the left-hand side of the decimal point, is 
a number less by i than the number of figures or places in the integer 
of the number. If a number contains both integers and decimals, the 
index is regulated according to the integers. If it contain only decimals, 
the index is equal to the number of cyphers next the decimal point, plus i; 
moreover, the index is negative, and is so distinguished by the sign minus, 
— , written over it 

For example, to illustrate the adjustment of the integer of the logarithm 
to the composition of the number : — 

Numbar. Logarithm. 

4743 3676053 

474.3 2.676053 

47.43 L676053 

4.743 0.676053 

.4743 .L676053 

.04743 .£.676053 

.004743 3676053 

Still more for the sake of compactness, the first two figures of the loga- 
rithms are given only at the beginning of each line of logarithms, to save 
repetition, only the remaining four decimal places being given for each 
logarithm. In seeking for a logarithm, the eye readily takes in the prefixed 
two digits at the commencement of each line. 

Rules, — To find the logarithm of a number containing one or two digits, 
look for the number in the preliminary tablet in one of the columns 
marked No., and find the logarithm next it Or, look in the body of 
the table for the given number in the columns marked N, with one or 
two cyphers following it; the decimal part of the logarithm is in the 
column next to it For example, the decimal part of the logarithm of 3 
is found, in the column next to the number 300, to be .477121, and as 
there is but one digit, the logarithm is completed with a cypher, thus, 
0.47 7 12 1. The same logarithm stands for 30, except that, when completed, 
it becomes 1.477 121. Again, take the number 37; look for 370 in column 
N, and the decimal part of the logarithm is found, in the colunm next it, 
to be .568202, which, being completed, becomes 1.568202. If the number 

be .37, the logarithm becomes 1.568202. 
To find the logarithm of a number consisting of three digits, look for the 



EXPLANATION AND USES OF THE TABLES. 33 

number in column N, and find the logarithm in the column next it, as 
already exemplified, for which the index is to be setded and prefixed as 
before. 

If the number consist of four digits, look for the first three in column N, 
and the fourth in the horizontal line at the head or at the foot of the table. 
The decimal part of the logarithm is found opposite the three first digits 
and under or over the fourth. Take the number 5432; opposite 543 in 
column N, and in the column headed 2, is the logarithm .734960, to which 
3 is to be prefixed, making 3.734960. If the number be 5.432, the 
complete logarithm is 0.734960. 

If the number consist of five or more digits, find the logarithm for the 
first four as above; multiply the difference, in column D, by the remaining 
digits, and divide by 10 if there be only one digit more, by 100 if there 
be two more, and so on; add the quotient to the logarithm for the first four. 
The sum is the decimal part of the required logarithm, to which the index 
is to be prefixed. For example, take 3. 141 6. The logarithm of 3 141 is 
.497068, decimal part; and the difference, 138 x 6 -h 10 = 83, is to be added, 
thus — 

0.497068 

83 



making the complete logarithm, 0.497151 

To find the number corresponding to a given logarithm, look for the 
logarithm without the index. If it be found exactly or within two or three 
units of the right-hand digit, then the first three figures of the indicated 
number will be found in the number column, in a line with the logarithm, 
and the fourth figure at the top or the foot of the column containing the 
logarithm. Annex the fourth figure to the first three, and place the decimal 
ix)int in its proper position, on the principles already explained. 

If the given logarithm differs by more than two or three units from the 
nearest in the table, find the number for the next less tabulated logarithm, 
which will give the four first digits of the required number. To find the 
fifth and sixth digits, subtract the tabulated logarithm from the given loga- 
rithm, add two c)rphers, and divide by the difference found in column D 
opposite the logarithm. Annex the quotient to the four digits already 
found,. and place the decimal point For example, to find the number 
represented by the logarithm 2.564732: — 

2.564732 given logaridim. 
Log. 367.0= 2.564666 nearest less. 



367.056 



56 D 118)6600 (56 nearly. 

590 



700 
708 



Showing that the required number is 367.056. 
To multiply together two or more numbers, add together the logarithms 

. 3 



34 MATHEMATICAL TABLES. 

of the numbers, and the sum is the logarithm of the product Thus, to 
multiply 365 by 3.146: — 

I-og 365 = 2.562293 

Log 3.146 = 0.497759 

3.060052 
Log 1148 3-059942 

29 D 380)11000 (29 nearly. 

760 

1148.29 

3400 

3420 

Showing that the product is 1148.29. 

To divide one number by another, subtract the logarithm of the divisor 
from that of the dividend, and the remainder is the logarithm of the 
quotient 

To find any power of a given number, multiply the logarithm of the num- 
ber by the exponent of the power. The product is the logarithm of the 
power. 

To find any root of a given number, divide the logarithm of the number 
by the index of the root The quotient is the logarithm of the root 

To find the reciprocal of a number, subtract the decimal part of the 
logarithm of the number from 0.000000; add i to the index of the loga- 
rithm, and change the sign of the index. This completes the logarithm of 
the reciprocal. For example, to find the reciprocal of 230: — 

0.000000 
Log 230 = 2.361728 

3.638272= log 0.004348 (reciprocal). 

Inversely, to find the reciprocal of the decimal .004348 :■— 

0.000000 
Log .004348 = 3.638272 

2.361728 = log 230 (reciprocal). 

Note. — It will be found in practice, for the most part, unnecessary to 
note the indices of logarithms, as the decimal parts are in most cases suffi- 
ciently indicative of the numbers without the indices. The exact calcula- 
tion of differences may also in most cases be dispensed with — rough mental 
approximations being sufficiently near for the purpose — particularly when 
the numbers contain decimals. The indices are, however, indispensable in 
the calculation of the roots of numbers. 



EXPLANATION AND USES OF THE TABLES. 35 

Tabl^ No. IL — Of Hyperbolic Logarithms of Numbers. 

In this table, the numbers range from i.oi to 30, advancing by .01, up 
to the whole number 10; and thence by larger intervals up to 30. The 
h)'perbolic logarithms of numbers, or Neperian logarithms, as they are 
sometimes called, are calculated by multiplying the common logarithms of 
the given numbers, in table No.. I., by the constant multiplier, 2.302585. 
The hyperbolic logarithms of numbers intermediate between those which 
are given in the table, may be readily obtained by interpolating proportional 
differences. 

Table No. III. — Of Circumferences, Circular Areas, Squares and 
Cubes; and of Square Roots and Cube Roots. 

It ha«g been shown how to calculate the powers and roots of numbers by 
means of logarithms. The table No. III. will be useful for reference. It 
contains the powers and roots of numbers consecutively from i to 1000. 
The circumferences and areas of circles, due to the numbers contained in 
the first columns, considered as diameters, are also given. They will be 
found useful when diameters are expressed in integers and decimals, or 
otherwise than in feet, inches, and fractions. 

Table No. IV. — Of Circumferences and Areas of Circles, with 

Sides of Equal Squares. 

The Table No. IV. gives the circumferences and areas of circles from 
^ inch to 120 inches in diameter, advancing by sixteenths of an inch up 
to 6 inches diameter; thence by eighths of an inch to 50 inches diameter; 
thence by quarters of an inch to 100 inches diameter; and thence by half 
inches to 120 inches diameter. 

At the same time, the decimal equivalents of fractions of inches are given 

in the first columns, and they are complemented by inches and decimals 

advancing by tenths, for which also the circumferences and areas are given. 

The table is thus completed for diameters expressed with decimals, as 

well as for those expressed with vulgar fractions. 

By a suitable adjustment of decimal points the circumferences and areas 
may be determined from the contents of the table for diameters ten or a 
hundred times as much as, or less than, the values given in the first column. 
^\Tiilst the diameters are here expressed as inches, they may be taken as 
feet, or as measures of any other denomination. 

The column of sides of equal squares^ contains the sides of squares having 
the same area as the circles in the same hnes of the table respectively. 

Note, — ^The column oi circular areas given in table No. III., contains the 
areas of circles of which the diameters are given in common numbers in 
the first column. 

Tables Nos. V. and VI. — Of Lengths of Circular Arcs. 

The lengths of circular arcs are given proportionally to that of the radius, 
and to that of the chord, in the tables Nos. V. and VL In the first of these 
tables, the radius is taken = i, and the number of degrees in the arc are 
given in the first column. The length of the arc as compared with the 
radius is given decimally in the second column. 



36 MATHEMATICAL TABLES. 

In the second table, the chord is taken = i, and the rise or height of the 
arc, expressed decimally as compared with the chord, is given in the first 
column. The length of the arc relatively to the chord is given in the 
second column. 

To use the first table, No. V., find the proportional length of the arc 
corresponding to the degrees in the arc, and multiply it by the actual length 
of the radius; the product is the actual length of the arc. 

To use the second table, No. VI., divide the height of the arc by the chord 
for the proportional height of the arc, which find in the first column of the 
table ; die proportional length of the arc corresponding to it being multi- 
plied by the actual length of the chord, gives the actual length of the arc. 

Note, — The length of an arc of a circle may be found nearly thus: — 
Subtract the chord of the whole arc from 8 times the chord of half the arc. 
A third of the remainder is the length nearly. 

Table No. VII. — Of Areas of Circular Segments. 

The areas of circular segments are given in Table No. VII., in proportional 
superficial measure, the diameter of the circle of which the segment forms 
a portion being = i. The height of the segment, expressed decimally in 
proportion to the diameter, is given in the first column, and the relative 
area in the second column. 

To use the table, divide the height by the diameter, find the quotient in 
the table, and multiply the corresponding area by the square of the actual 
length of the diameter; the product will be the actual area. 

Table No. VIII. — Sines, Cosines, Tangents, Cotangents, Secants, 

AND Cosecants of Angles from o° to 90° 

This table, Na VIII., is constructed for angles of from 0° to 90°, advancing 
by 10', or one-sixth of a degree. The length of the radius is equal to i, 
and forms the basis for the relative lengths given in the table, and which 
are given to six places of decimals. Each entry in the table has a duplicate 
significance, being the sine, tangent, or secant of one angle, and at the 
same time the cosine, cotangent, or cosecant of its complement For this 
reason, and for the sake of compactness, the headings of the columns are 
reversed at the foot; so that the upper headings are correct for the angles 
named in the left hand mai^n of the table, and the lower headings for 
those named in the right hand margin. 

To find the sine, or other element, to odd minutes, divide the difference 
between the sines, &c., of the two angles greater and less than the given 
angle, in the same proportion that the given angle divides the difference of 
the two angles, and add one of the parts to the sine next it 

By an inverse process the angle may be found for an^ given sine, &c., 
not found in the table. 

Table No. IX. — Of Logarithmic Sines, Cosines, Tangents, and Co- 
tangents of Angles from 0° to 90^ 

This table. No. IX., is constructed similarly to the table of natural sines, 
&c., preceding. To avoid the use of logarithms with negative indices, the 
radius is assumed, instead of being equal to i, to be equal to 10", or 



EXPLANATION AND USES OF THE TABLES. 37 

io,cx)o,ooo,ooo; consequently the logarithm of the radius = lo log lo = lo, 
^^Tience, if, to log sine of any angle, when calculated for a radius = i, there 
be added lo, the sum will be the log sine of that angle for a radius = lo". 
For example, to find the logarithmic sine of the angle 15° 50'. 

Nat sine 15'' 50'= -272840; its log = 1*435908 

add = 10 



Logarithmic sine of 15** 50'= 9*435908 

When the logarithmic sines and cosines have been found in this manner, 
the logarithmic tangents, cotangents, secants, and cosecants are found from 
those by addition or subtraction, according to the correlations of the 
trigdnometrical elements already given, and here repeated in logarithmic 

fonn: — 

Log tan = 10 + log sin. - log. cosin. 

Log cotan = 20 — log tan. 

Log sec. = 20 — log cosin. 

Log cosec = 20 - log sin. 

To find the logarithmic sine^ tangent^ &*c., of any angle, — ^When the number 
of degrees is less than 45°, find the degrees and minutes in the left hand 
colunm headed angle^ and under the heading sine^ or tangent^ &c., as 
required, the logarithm is found in a line with the angle. 

When the number of degrees is above 45°, and less than 90®, find the 
degrees and minutes in the right hand column headed angle^ and in the 
same line, above the title at the foot of the page, sine or tangent^ &c., find 
the logarithm in a line with the angle. 

When the number of degrees is between 90° and 180°, take their supple- 
ment to 180°; when between 180° and 270°, diminish them by 180°; and 
when between 270° and 360°, take their complement to 360°, and find the 
logarithm of the remainder as before. 

If the exact number of minutes is not found in the table, the logarithm 
of the nearest tabular angle is to be taken and increased or diminished as 
the case may be, by the due proportion of the difference of the logarithms 
of the angles greater and less than the given angle. 

Table No. X. — Rhumbs, or Points of the Compass. 

The Mariner's Compass is a circular card suspended horizontally, having 
a thin bar of steel magnetized, — the needle^ — for one of its diameters; the 
circumference of the card being divided into 32 equal parts, ox points^ and 
each point subdivided into quarters. A point of the compass is, therefore, 
equal to (360** ^ 32 = ) n° 15'. 

Table No. XL — Of Reciprocals of Numbers. 

The table No. XL contains the reciprocals of numbers from i to 1000. 
It has already been shown how to find the reciprocal of a number by means 
of logarithms. 



38 



MATHEMATICAL TABLES. 



TABLE No. L— LOGARITHMS OF NUMBERS 

FROM I TO 10,000. 



No. 


Log. 


1^0. 


Log. 


No. 


Log. 


No. 


Log. 


1 


o.oooooo 


26 


1.414973 


51 


1.707570 


76 


1. 880814 


2 


0.301030 


27 


1.431364 


52 


1. 716003 


77 


1 1. 88649 1 


3 


0.477I2I 


28 


. 1.447158 


53 


1.724276 


78 


1.892095 


4 


0.602060 


29 


1.462398 


54 


1.732394 


79 


1.897627 


5 


0.698970 


30 

1 


1.477121 


55 


1.740363 


80 


1.903090 


6 


0.778I5I 


31 


1.491362 


56 


I. 748188 


81 


1.908485 


7 


0.845098 


32 


1.505150 


57 


1-755875 


82 


1.913814 


8 


0.903090 


33 


1.518514 


5« 


1.76342S 


P 


1. 919078 


9 


0954243 


34 


1.531479 


59 


1.770852 


84 


1.924279 


lO 


I.OOOOOO 


35 


1.544068 


60 


1.77S151 


85 


1.929419 


11 


I.04I393 


36 


1-556303 


61 


1.785330 


86 


1.934498 


12 


I.079I8I 


37 


1.568202 


62 


1.792392 


87 


1.939519 


13 


1. 1 13943 


38 


1.579784 


63 


I. 799341 


88 


1.944483 


H 


I.I46I28 


39 


1. 591065 


64 


1. 8061 80 


89 


I -949390 


15 


I.I7609I 


40 


1.602060 


65 


1.812913 


90 
91 


1.954243 


i6 


1. 204120 


41 


1. 612784 


66 


1-819544 


1.959041 


'Z 


1.230449 


42 


1.623249 


^7 


1.826075 


92 


1.963788 


i8 


1-255273 


43 


1.633468 


68 


1.832509 


93 


1.968483 


19 


1.278754 


44 


1.643453 


69 


1.838849 


94 


1.973128 1 


20 


I. 301030 


45 


1. 653213 


70 


1.845098 
1.851258 


95 


1.977724 


21 


1. 322219 


46 


1.662758 


71 


96 


1. 98227 1 


22 


1.342423 


47 


1.672098 


72 


1.857332 


97 


1.986772 


23 


1. 361728 


48 


I.68i24< 


73 


1.863323 


98 


1. 991 226 


24 


1. 38021 1 


49 


1. 690196 


74 


1.869232 


99 


1.995635 


25 


I -397940 


50 


1.698970 


75 


I. 875061 


100 


2.000000 



N 





I 


2 


3 


4 


5 


— 1 

6 


7 


8 


9 


D 

432 


100 


00- 0000 


0434 


0868 


1301 


1734 


2166 


2598 


3029 


3461 


3891 


lOI 


00- 4321 


4751 


51S1 


5609 


6038 


6466 


6894 


7321 


7748 


8174 


428 


102 


00- 8600 


9026 


9451 


9876 














425 


102 


01- 








0300 


0724 


1 147 


1570 


1993 


2415 


424 


103 


01- 2837 


3259 


3680 


4100 


4521 


4940 


5360 


5779 


6197 


6616 1 420 


104 


01- 7033 


7451 


7868 


8284 


8700 


9116 


9532 


9947 




1 417 


104 


02- 
















0361 


0775 416 


105 


02- I I 89 


1603 


2016 


2428 


2841 


3252 


3664 


4075 


4486 


4896 412 


106 


02- 5306 


5715 


6125 


6533 


6942 


7350 


7757 


8164 


8571 


8978 


408 


107 


02- 9384 


9789 


















405 


107 


03- 




0195 


0600 


1004 


1408 


1812 


2216 


2619 


3021 


404 


108 


03- 3424 


3826 


4227 


4628 


5029 


5430 


5830 


6230 


6629 


7028 


400 


109 


03- 7426 


7825 


8223 


8620 


9017 


9414 


981 1 








398 


109 

N 


04- 














0207 


0602 
8 


0998 
9 


397 





I 


2 


3 


4 


5 


6 


7 


D 



LOGARITHMS OF NUMBERS. 



39 



N 





I 
1787 


2 
2182 


3 
2576 


4 


5 


6 


7 


8 


9 


D 


110 i 04- 1393 


2969 


3362 


3755 


4148 


4540 


iP^ 


393 


! 


'" ' 04- 5323 


5714 


6105 


6495 


6885 


7275 


7664 


8053 


8442 


8830 


389 




112 04- 9218 


9606 


9993 
















388 




112 05- 






0380 


0766 


1153 


1538 


1924 


2309 


2694 


386 




"3 05- 3078 


3463 


3846 


4230 


4613 


4996 


5378 


5760 


6142 


6524 


383 


114 1 05- 6905 


7286 


7666 


8046 


8426 


8805 


9185 


9563 


9942 




383 




114 06- 


















0320 


379 


1 

 


L15 06- 0698 


1075 


1452 


1829 


2206 


2582 


2958 


3333 


3709 


4083 


376 




116 06- 4458 

117 06- 8186 


4832 


5206 


5580 


5953 


6326 


6699 


7071 


7443 


7815 


373 




8557 


8927 


9298 


9668 




• 








380 




"7 07- 










0038 


0407 


0776 


1 145 


1514 


370 




118 07- 1882 


2250 


2617 


2985 


3352 


3718 


4085 


4451 


4816 


5182 


366 




119 07- 5547 


5912 


6276 


6640 


7004 


7368 


773» 


8094 


8457 


8819 


363 




120 1 07- 9181 


9543 


9904 


• • * • • 














362 




120 


08- 






0266 


0626 


0987 


1347 


1707 


2067 


2426 


360 




121 


08- 2785 


3^44 


3503 


3861 


4219 


^576 


4934 


5291 


5647 


6004 


357 


1 122 


08- 6360 


6716 


7071 


7426 


7781 


8136 


8490 


8845 


9198 


9552 


355 


: 123 08- 9905 




















355 


123 09- 


0258 


0611 


0963 


'3'5 


1667 


2018 


2370 


2721 


3071 


353 


1 


124 09- 3422 


3772 


4122 


4471 


4820 


5169 


5518 


5866 


6215 


6562 


349 


125 


09- 6910 


7257 


7604 


795 » 


8298 


8644 


8990 


9335 


9681 




348 




125 


10- 


















0026 


346 




126 


10- 0371 


0715 


1059 


1403 
482! 


1747 


2091 


2434 


2777 


3"9 


3462 


343 




»27 


xo- 3804 


4146 


4487 


5169 


5510 


5851 


6191 


6531 


6871 


341 


128 


10- 7210 


7549 


7888 


8227 


8565 


8903 


9241 


9579 


9916 




338 


12S ] 


11- 


















0253 


337 




129 


XI- 0590 


0926 


1263 


1599 


1934 


2270 


2605 


2940 


3275 


3609 


335 


130 


XI- 3943 


4277 


461 1 


4944 


5278 
8595 


5611 


5943 


6276 


6608 


6940 


333 


131 


IX- 7271 


7603 


7934 


8265 


8926' 


9256 


9586 


9915 




331 


131 12- 


















0245 


330 


132 i 12- 0574 


0903 
4178 


1231 


1560 


1888 


2216 


2544 
5806 


2871 


3198 


3525 


328 


m , "- 3852 


4504 


4830 


5156 
8399 


5481 


6131 


6456 


6781 


325 


134 12- 7105 


7429 


7753 


8076 


8722 


9045 


9368 


9O90 




323 




1 *34 . 13- 


















0012 


323 


18« 


13- 0334 


o6« 


0977 


1298 


1619 


1939 


2260 


2580 


2900 


3219 


321 


'136 


13- 3539 


3858' 


4177 


4496 


4814 


5133 


5451 


5769 


6086 


6403 


318 


!i37 


X3- 6721 


7037 


7354 


7671 


7987 


8303 


8618 


8934 


9249 


9564 


316 




'^0 


13- 9879 




















315 




138 


14- 


0194 


0508 


0822 


1 136 


1450 


1763 


2076 


2389 


2702 


3H 




139 


14- 3o»5 


3327 


3639 


3951 


4263 


4574 


4885 


5196 


5507 


S8i8 


3" 


140 14- 6128 


6438 


6748 


7058 


7367 


7676 


7985 


8294 


8603 


8911 


309 


141 


14- 9219 


9527 


9835 
















308 


:Hi 


^5- •- • 






0142 


0449 


0756 


1063 


1370 


1676 


1982 


307 


142 


X5- 2288 


2594 


2900 


3205 


3510 


lS'5 


4120 


4424 


4728 


5032 
8061 


305 


143 


15- 5336 


5640 
8664 


5943 
8965 


6246 


6549 


6852 


7154 


7457 


7759 


303 




144 


15- 8362 


9266 


9567 


9868 






• • • • • • 




302 




144 


16- 












0168 


0469 


0769 


1068 


301 




145 


I 16- 1368 


1667 


1967 


2266 


2564 


2863 


3161 


3460 


3758 


4055 


299 


M46 1 16- 4353 


4650 


4947 


5244 


5541 


5838 


6134 


6430 
9380 


6726 


7022 


297 




>47 


i^ 7317 


7613 


7908 8203 


8497 


8792 


9086 


9674 


9968 


295 
D 




N) 





I 


2 


3 


4 


5 


6 


7 


8 


9 



40 



MATHEMATICAL TABLES. 



N 
148 





I 


2 


3 


4 


5 


6 


7 


8 


9 


D 

293 


17- 0262 


0555 


0848 


II4I 


1434 


1726 


2019 


2311 


2603 


2895 


149 


17- 3186 


3478 


3769 


4060 


4351 


4641 


4932 


5222 


S5I2 


5802 


291 


150 


17- 6091 


6381 


6670 


6959 


7248 


7536 


7825 


8113 


8401 


8689 


289 


151 


17- 8977 


9264 


9552 


9839 














287 


151 


18- 








0126 


0413 


0699 


0986 


1272 


1558 


287 


IS2 


z8- 1844 


2129 


2415 


2700 


2985 


3270 


3555 


3839 


4123 


4407 


^?5 


I S3 


18- 4691 


4975 


5259 


5542 


5825 


6108 


6391 


6674 


6956 


7239 


283 


154 


i8~ 7521 


7803 


8084 


8366 


8647 


8928 


9209 


9490 


9771 




281 


»54 


19- 


















0051 


281 
279 


155 


19- 0332 


0612 


0892 


II7I 


145 1 


1730 


2010 


2289 


2567 


2846 


156 


19- 3125 


3403 


3681 


3959 


4237 


45 »4 


4792 


5069 


5346 


5623 


278 


«57 


19- 5900 


6176 


6453 


6729 


7«>5 


7281 


7556 


7832 


8107 


8382 


276 


'55 


19- 8657 


8932 


9206 


9481 


9755 












275 


158 


2(>- 










0029 


0303 


0577 


0850 


1124 


274 


159 


20- 1397 


1670 


»943 


2216 


2488 


2761 


3033 


3305 


3577 


3848 


272 


160 


20> 4120 


4391 


4663 


4934 


5204 


5475 
8173 


5746 
E441 


6016 


6286 


6556 


271 


161 


20- 6826 


7096 


7365 


7634 


7904 


8710 


8979 


9247 


269 


162 


20- 9515 


9783 


















268 


162 


21- 




c»5i 


0319 


0586 


0853 


1121 


1388 


1654 


1921 


267 


163 


21- 2188 


2454 


2720 


2986 


3252 


3518 


3783 


4049 


43H 


4579 


266 


164 


21- 4844 


5109 


5373 


5638 


5902 


6166 


6430 


6694 


6957 
9585 


7221 


264 


166 


21- 7484 


7747 


8010 


8273 


8536 


8798 


9060 


9323 


9846 


262 


166 


22- 0108 


0370 


0631 


0892 


"53 


1414 


1675 


1936 


2196 


2456 


261 


167 


22- 2716 


2976 


3236 


3496 


3755 


4015 


4274 


4533 


4792 


5051 


259 


168 


22- 5309 


5568 


5826 


6084 


6342 


6600 


6858 


7115 


7372 


7630 


258 


169 


22- 7887 


8144 


5400 


8657 


8913 


9170 


9426 


9682 


9938 




257 


169 


23- 


















0193 256 


170 


23- 0449 


0704 


0960 


1215 


1470 


1724 


1979 


2234 


2488 


2742 


255 


171 


23- 2996 


3250 


3504 


3757 


401 1 


4264 


4517 


4770 


5023 


5276 


253 


172 


23- 5528 
23- 8046 


5781 


S^33 


6285 


6537 


6789 


7041 


7292 


7544 


7795 


252 


173 


8297 


8548 


8799 


9049 


9299 


9550 


9800 






251 


173 


24- 














 • «  B  


0050 


0300 


250 


174 


24- 0549 


0799 


1048 


1297 


1546 179s 


2044 


2293 


2541 


2790 


249 


175 


24- 3038 


3286 


3534 


3782 


4030 


4277 


4525 


4772 


5019 


5266 


248 


176 


24- 55^3 


5759 
8219 


6006 


6252 


6499 


6745 


6991 


72J7 


7482 


7728 


246 


177 


24- 7973 


8464 


8709 


8954 


9198 


9443 


9687 


9932 




246 


177 


25- 


















0176 


245 


178 


25- 0420 
25- 2853 


0664 


o9oi; 


3580 


1395 


1638 


1881 


2121 
4548 


2368 


2610 


243 


179 


3096 


3338 


3822 


4064 


4306 


4790 


5031 


242 


180 


25- 5273 


5514 


5755 


5996 


6237 


6477 


6718 


6958 


7198 


7439 


241 


181 


25- 7679 


7918 


8158 


8398 


8637 


8877 


9116 


9355 


9594 


9833 


239 


182 


26- 0071 


0310 


0548 


0787 


1025 


1263 


1 501 


1739 


1976 


2214 


238 


'S3 


26- 2451 


2688 


2925 


3162 


3399 


3636 


3873 


4109 


4346 


4582 


237 


184 


26- 4818 


5054 


5290 


5525 


5761 


5996 


6232 


6467 


6702 


6937 


235 


185 


26- 7172 


7406 


7641 


7875 


81 10 


8344 


8578 


8812 


9046 


9279 


234 


186 


26- 9513 


9746 


9980 
















234 


186 


27- 






0213 


0446 


0679 


0912 


1144 


1377 


1609 


233 


'SZ 


27- 1842 


2074 


2306 


2538 
4850 


2770 


3001 


3233 


3464 


3696 


3927 


232 


188 


27- 4158 


4389 


4620 


5081 


53" 


5542 


5772 


6002 


6232 


230 
D 


N 





I 


2 


3 


4 


5 


6 


7 


8 


9 



LOGARITHMS OF NUMBERS. 



41 



N 1 o 


I 


2 


3 


4 


5 


6 


7 


8 


9 


D 
229 

228 




189 27- 6462 


6692 


6921 


7151 


7380 7609 


7838 


8067 


8296 


8525 




190 ay- 8754 


8982 


921 1 


9439 


9667 


9895 












190 a8- 










a • • •  « 


0123 


035' 


0578 
2849 


0806 


228 




191 a8- 1033 


1261 


1488 


1715 


1942 


2169 


2396 


2622 


3075 


227 




192 


aa- 3301 


3527 


3753 


3979 


4205 


4431 


4656 


4882 


5107 


5332 


226 




193 


a8- SSS7 


i^6 


6007 


6232 


6456 


6681 


6905 


7130 


7354 
9589 


7578 
9812 


225 




194 


a8- 7802 


8249 


8473 


8696 


8920 


9143 


9366 


223 




195 


29- 0035 


0257 


Q480 


0702 


0925 


1 147 


1369 


1591 
3804 


1813 


2034 


222 




196 


29- 2256 


2478 


2699 


2920 


3141 


3363 


3584 


4025 


4246 


221 




'^Z 


ag- 4466 


4687 


4907 


5127 


5347 


5567 


5787 


6007 


6226 


6446 


220 




198 


39- 6665 


6884 


7104 


7323 


7542 


7761 


7979 


8198 


8416 


8635 


219 




X99 


29-8853 


9071 


9289 


9507 


9725 


9943 










218 




'99 


30- 












0161 


0378 


0595 


0813 


2x8 




200 


30- 1030 


1247 


1464 


1681 


180R 


2114 


233« 


2547 


2764 


2980 


217 




201 


30- 3196 


3412 


3628 


3844 


4059 


4275 


4491 


4706 


4921 


5136 


216 




202 


30- 535 > 


5566 


5781 


5996 
8137 


621 1 


6425 


6639 


6854 


7068 


7282 


215 




203 


30- 7496 


7710 


7924 


8351 


8564 


8778 


8991 


9204 


9417 


213 




204 


30- 9630 


9843 






• a  • • 










• «  •  


213 




204 


3X- 




0056 


0268 


0481 


0693 


0906 


1118 


1330 


1542 


212 




1205 


31- 1754 


1966 


2177 


2389 


2600 


2812 


3023 


3234 


3445 


3656 


211 




! 206 


31- 3867 


4078 


4289 


4499 


4710 


4920 


5130 


5340 


5551 


5760 


210 




207 


31- 5970 
31- 8063 


6180 


6390 


6599 


6809 


7018 


7227 


7436 


7646 


7854 


209 




208 


8272 


8481 


8689 


8898 


9106 


9314 


9522 


9730 


9938 


208 




209 

210 


32- 0146 


0354 


0562 


0769 


0977 


1 184 


1391 


1598 


1805 


2012 


207 




32- 2219 


2426 


2633 


2839 


3046 


3252 


3458 


3665 


3871 


4077 


206 




211 


32- 4282 


44^ ;8 


4694 


4899 


5105 


5310 


5516 


5721 


5926 


6131 


205 




212 


32- 6336 
32- 8380 


6541 


6745 


6950 


7155 


7359 


7563 


7767 


7972 


8176 


204 




213 


8583 


8787 


8991 


9194 


9398 


9601 


9805 






204 




2J3 


33- 
















0008 


0211 


203 




2H 


33- 0414 


0617 


G819 


1022 


1225 


1427 


1630 


1832 


2034 


2236 


202 




215 


33- 2438 


2640 


2842 


3044 


3246 


3447 


3649 


3850 


4051 


4253 


202 




216 


33- 4454 


4655 


4856 


5057 


5257 


5458 


5658 


5859 


6059 


6260 


201 




217 


33- 6460 


6660 


6860 


7060 


7260 


7459 


7659 


7858 


«os« 


8257 


200 


m 


218 


33- 8456 


8656 


8855 


9054 


9253 


9451 


9650 


9849 






200 




218 


34- 














  • • • 


0047 


0246 


199 




219 34- 0444 


0642 


0841 


1039 


1237 


1435 


1632 


1830 


2028 


2225 


198 




220 


34- 2423 


2620 


2817 


3014 


3212 


3409 


3606 


3802 


3999 


4196 


197 




. 221 


34- 4392 


4589 


4785 


4981 


5178 


5374 


5570 


5766 


5962 


6157 


196 




222 


34- 6353 


6549 


6744 


6939 


7135 


7330 


7525 


7720 


7915 


8110 


195 




223 


34- 8305 


8500 


8694 


8889 


9083 


9278 


9472 


9666 


9860 




194 




223 


35- 










 « •  • 








0054 
1989 


194 




224 


35- Q248 


0442 


c6i6 


0829 


1023 


1216 


1410 


1603 


1796 


193 




'225 35- 21^3 


2375 


2568 


2761 


4876 


3147 


3339 


3532 


3724 


3916 


193 


1 226 


35- 4108 


4301 


4493 
6408 


4685 


5068 


5260 


5452 


5643 


5834 


192 




227 


35- 6026 


6217 


6599 


6790 


6981 


7172 


7363 


7554 


7744 


191 


1 


228 


35- 7935 


8125 


8316 


8506 


86q6 


8886 


9076 


9266 


9456 


9646 


190 


1 


229 


35-9835 




















189 


1 

1 


229 


35- 


0025 


0215 


0404 


0593 


0783 


0972 


1 161 


1350 


1539 


189 
D 




N 





I 


2 


3 


4 


5 


6 


7 


8 


9 


i 



42 



MATHEMATICAL TABLES. 



N 



6 



8 



D 



230 36- 1728 191 7 2105 2294 2482 

231 36-3612 3800 3988 4176 4363 

232 36- 5488 5675 5862 6049 6236 

233 36- 7356 7542 7729 7915 8101 

234 36- 9216 9401 9587 9772 9958 



2671 2859 3048 

4551 4739 4926 

6423 6610 6796 

8287 8473 8659 



3236 3424 

5"3 5301 

6983 7169 

8S45 9030 



P •  a • 



[88 
[88 
187 
[86 
[86 



234 

235 
236 

237 
238 

239 
239 



37- 

37- 1068 
37- 2912 
37- 4748 
37- 6577 

37- 8398 

38- 



1253 1437 1622 1806 

3096 3280 3464 3647 

4932 5"5 5298 5481 

6759 6942 7124 7306 

8580 8761 8943 9124 



0143 
1991 

3831 
5664 

7488 

9306 



0328 
2175 

4015 
5846 

7670 
9487 



05 '3 
2360 

4198 

6029 

7852 
9668 



0698 

2544 
4382 
6212 
8034 

9849 



0883 
2728 

4565 

6394 
8216 

0030 



(85 
[84 
[84 

^83 

[S2 
[82 
[81 



240 

241 
242 

243 
244 



38- 021 I 
38- 2017 

38- 3815 
38- 5606 
38- 7390 



0392 
2197 

3995 
578§ 
7568 



0573 
2377 
4174 
5964 
7746 



0754 

2557 

4353 
6142 

7923 



0934 

2737 

4533 
6321 

8101 



I I 15 1296 1476 

2917 3097 3277 

4712 4891 5070 

6499 6677 6856 

8279 8456 8634 



1656 1837 181 

3456 3636 180 

5249 5428 179 

7034 7212 178 

8S11 8989 178 



245 

245 
246 

247 

248 

249 



38- 9166 

39- 

39- 0935 
39- 2697 
39- 4452 
39- 6199 



9343 9520 9698 9875 



1112 

2873 
4627 

6374 



1288 
3048 
4802 
6548 



1464 
3224 

4977 
6722 



1641 
3400 
5152 
6896 



0051 
1817 

3575 
5326 
7071 



0228 

1993 
3751 
5501 

724s 



0405 
2169 
3926 
5676 

7419 



0582 

2345 
4101 

5850 
7592 



0759 
2521 

4277 
6025 

7766 



77 

77 
76 

76 

75 
74 



250 

251 
251 
252 

253 
254 

255 

256 

257 

257 
258 

259 



39- 7940 

39- 9674 

40- 

40- 1401 
40- 3121 
40- 4834 



8114 
9847 

1573 
3292 
5cx>s 



8287 8461 8634 



0020 

1745 
3464 
S176 



0192 
1917 

3635 
5346 



0365 
2089 

3807 

5517 



8808 8981 9154 9328 9501 



0538 
2261 

3978 
$688 



071 1 

2433 
4149 

5858 



0883 
2605 
4320 
6029 



1056 
2777 

4492 
6199 



1228 

2949 
4663 

6370 



73 
73 
73 
72 
71 
71 



40- 6540 
40- 8240 

40- 9933 

41- 

41- 1620 
41- 3300 



6710 
8410 

0102 
1788 

3467 



6881 
8579 

0271 
1956 

3635 



7051 
8749 

• • • • • 

0440 
2124 

3803 



7221 
8918 

0609 
2293 
3970 



739X 
9087 

0777 
2461 

4137 



7561 
9257 

 •   • 

0946 
2629 

4305 



7731 
9426 

1114 
2796 
4472 



7901 
9595 

1283 
2964 

4639 



8070 
9764 

1451 

3>32 
4806 



70 
[69 
[69 
[69 
[68 

67 



260 

261 
262 
263 
263 
264 



41- 4973 
41- 6641 

41- 8301 

41- 9956 

42- 

42- 1604 



5140 
6807 

8467 

0121 
1768 



5307 
6973 
8633 

0286 
1933 



5474 
7139 
8798 

0451 
2097 



5641 ] 5808 

7306 

8964 



0616 
2261 



7472 
9129 

0781 
2426 



5974 
7638 

9295 

0945 
2590 



6141 

7804 
9460 

1 110 
2754 



6308 
7970 
9625 

1275 
2915 



6474 

8135 
9791 

1439 
3082 



[67 
[66 
[6s 
[65 
[65 
[64 



265 

266 
267 
268 
269 
269 



270 

271 

N 



42- 3246 
42- 4882 
42- 651 I 
42- 8135 

42- 9752 

43- 



3410 

5045 
6674 

8297 

9914 



3574 
5208 

6836 
8459 



3737 

5371 
6999 

8621 



3901 

5534 
7161 

8783 



0075 0236 0398 



4065 

5697 

7324 
8944 



4228 
5860 
7486 
9106 



4392 
6023 
7648 
9268 



4555 
6186 

781 1 

9429 



4718 
6349 
7973 
9591 



0559 0720 0881 1042 1203 



43- 1364 
43- 2969 



1525 
3130 



1685 
3290 



1846 
3450 



2007 
3610 



2167 
3770 



2328 
3930 



2488 
4090 

7 



2649 
4249 

8 



2809 
4409 



[64 
163 

t62 

[62 
[62 
[61 



[61 

60 

D 



LOGARITHMS OF NUMBERS. 



43 



N 





I 


2 


3 


4 


5 


6 


7 


8 


9 


D 


272 


43- 45^ 


4729 


4888 


5048 


5207 


5367 


5526 


5685 


5844 


6004 


VS9 


273 


43- 6163 


6322 


64S1 


6640 


6799 


6957 


7116 


7275 


7433 


7592 


159 


274 


43- 7751 


7909 


8067 


8226 


8384 


8542 


8701 


8859 


9017 


9175 


158 


275 


43- 9333 


9491 


9648 


9806 


9964 












158 


275 


44- 










0122 


0279 


0437 


0594 


0752 


158 


276 


44- 09C39 


1066 


1224 


1381 


iS3^ 


1695 


1852 


2009 


2166 


2323 


157 


277 


44- 2480 


2637 


2793 


2950 


3106 


3263 


3419 


3576 


3732 


3889 


157 


278 


44- 4045 


4201 


4357 


4513 


4669 


4825 


4981 


5137 


5293 


5449 


156 


279 


44- 5604 


5760 


5915 


6071 


6226 


6382 


6537 


6692 


6848 


7003 


155 


280 


44- 7158 


Z3'3 


7468 


7623 


7778 


7933 


8088 


8242 


8397 


8552 


155 


281 


44- 8706 


8S61 


9015 


9170 


9324 


9478- 


9633 


9787 


9941 




154 


281 


45- 


















0095 


154 


282 


45- 0249 


0403 


0557 


071 1 


0865 


1018 


1 172 


1326 


1479 


1633 


154 


^S^ 


45- 1786 


1940 


2093 


2247 


2400 


2553 


2706 


2859 


3012 


3165 


153 


284 


45- 3318 


3471 


3624 


3777 


3930 


4082 


4235 


4387 


4540 


4692 


153 


285 


45- 4845 


4997 


5150 


5302 


5454 


5606 


5758 


5910 


6062 


6214 


^52 


286 


45- 6366 


6518 


6670 


6821 


6973 


7125 


7276 


7428 


7579 


7731 


152 


287 


45- 7882 


«033 


8184 


8336 


8487 


8638 


8789 


8940 


9091 


9242 


151 


288 


45- 9392 


9543 


9694 


9845 


9995 












151 


2SS 


46- ...... 










0146 


0296 


0447 


0597 


0748 


151 


289 


46- 0S98 


1048 


1198 


1348 


1499 


1649 


1799 


1948 


2098 


2248 


150 


280 


46- 2398 


2548 


2697 


2847 


2997 


3146 


3296 


3445 


3594 


3744 


150 


291 


46- 3893 


4042 


4191 


4340 


4490 


4639 


4788 


4936 


5085 


5234 


149 


292 


46- 5383 
46- 6868 


5532 


5680 


5829 


5977 


6126 


6274 


6423 


6571 


6719 


149 


293 


7016 


7164 


7312 


7460 


7608 


7756 


7904 


8052 


8200 


148 


294 


46- 8347 


8495 
9969 


8643 


8790 


8938 


9085 


9233 


9380 


9527 


9675 


148 


295 


45- 9822 


• B   • 














147 


295 


47- 




01 16 


0263 


0410 


0557 


0704 


0851 


0998 


1 145 


147 


296 


47- 1292 


1438 


I5«5 


1732 


1878 


2025 


2171 


2318 


2464 


2610 


146 


297 


47- 2756 


2903 


3049 


3195 


3341 


3487 


3633 


3779 


3925 


4071 


146 


298 


47- 4216 


4362 


4S08 


4653 


4799 


4944 


5090 


5235 


5381 


5526 


146 


299 


47- 5671 


5816 


5962 


6107 


6252 


6397 


6542 


6687 


6832 


6976 


145 


300 


47- 7121 


7266 


741 1 


7555 


7700 


7844 


7989 


8133 


8278 


8422 


145 


301 


47- 8566 


871 1 


8855 


8999 


9143 


9287 


9431 


9575 


9719 


9863 


144 


3«>2 


48— 0007 


0151 
1586 


0294 


0438 


0582 


0725 


0869 


1012 


"§S 


1299 


144 


393 


48- 1443 


1729 


1872 


2016 


2K9 


2302 


2445 


2588 


2731 


143 


304 


48- 2874 


3016 


3^59 


3302 


3445 


3587 


3730 


3872 


4015 


4157 


M3 


305 


48- 4300 


4442 


4585 


4727 


4869 


5011 


5153 


5295 


5437 


5579 


142 


306 


43- 5721 


5863 


6005 


6147 


6289 


6430 


6572 


6714 


6855 


6997 


142 


3^ 


48- 7138 


7280 


7421 


7563 


7704 


7845 


7986 


8127 


8269 


8410 


141 


SoS 


48- 8551 


8692 


8833 


8974 


9114 


9255 


9396 


9537 


9677 


9818 


141 


3P9 


48- 9958 




















140 


. 3P9 


49- 


0099 


0239 


0380 


0520 


0661 


0801 


0941 


108 1 


1222 


140 


310 


49- 1362 


1502 


1642 


1782 


1922 


2062 


2201 


2341 


2481 


2621 


140 


3^1 


49- 2760 


29CX> 


3040 


3179 


3319 


3458 


3597 


3737 


3876 


4015 


139 


312 


49- 4155 


4294 


4433 


4572 


471 1 


4850 


4989 


5128 


5267 


5406 


139 


313 


49- 5544 


5683 


5822 


5960 


6099 


6238 


6376 


6515 


6653 


6791 


139 


' 3«4 


49- 6930 


7068 


7206 


7344 


7483 


7621 


7759 


7897 


8035 


8173 


138 


N 





I 


2 


3 


4 


5 


6 


7 


8 


9 


D 



44 



MATHEMATICAL TABLES. 



N 



315 

316 
316 

317 
318 
319 



49- 83" 

49- 9687 

50- 1059 
50- 2427 

50- 3791 



8448 
9824 

1 196 
2564 

3927 



8586 
9962 

1333 
2700 

4063 



8724 8862 



• ••  • 

0099 
1470 

2837 
4199 



8 



8999 9137 9275 9412 9550 



0236 0374 
1607 1744 



2973 
4335 



3109 
4471 



0511 
1880 

3246 
4607 



0648 
2017 
3382 
4743 



0785 
2154 
35^8 
4878 



0922 
2291 

36SS 
5014 



D 

138 
^37 
t37 
[37 
^36 
^36 



320 

321 
322 

323 
323 
324 



50- 5150 
50- 6505 

50- 7856 

50- 9203 

51- 

51- 0545 



5286 
6640 
7991 

9337 



5421 
6776 
8126 

9471 



5557 
691 1 

8260 

9606 



5693 
7046 

8395 
9740 



0679 0813 0947 1081 



5828 
7181 

8530 
9874 

1215 



5964 
7316 
8664 

0009 
1349 



6099 

7451 
8799 

0143 
1482 



6234 
7586 

8934 

0277 
1616 



6370 
7721 

9068 

041 1 
1750 



^36 
135 
^35 
34 
'34 
134 



325 

326 

327 
328 

329 ' 



51- i88j 
51- 3218 
51- 4548 

51- 5874 
51- 7196 



2017 

3351 
4681 

6006 
7328 



2151 

3484 

4813 

6139 
7460 



2284 

3617 
4946 

6271 
7592 



2418 

3750 

5079 
6403 

7724 



2551 

3883 
521 1 

6535 
785s 



2684 
4016 

5344 
6668 

7987 



2818 
4149 

5476 
6800 
81 19 



2951 
4282 

5609 
6932 

8251 



3084 
4415 

5741 
7064 

8382 



^33 

'33 
133 
[32 
'32 



380 

331 
331 
332 
333 
334 



51- 8514 

51- 9828 

52- 

52- 1138 
52- 2444 

52- 3746 



8646 
9959 

1269 

2575 
3876 



8777 8909 9040 



0090 
1400 
2705 
4006 



022 X 
IJ30 

2835 
4136 



0353 
1661 

2966 

4266 



9171 9303 9434 9566 9697 



0484 
1792 
3096 
4396 



0615 
1922 
3226 
4526 



0745 
2053 
3356 
4656 



0876 
2183 
3486 

4785 



1007 

2314 
3616 

4915 



'31 
31 
'31 
31 
[30 

'30 



385 

336 

337 
338 
338 
339 



52- 5045 
52- 6339 
52- 7630 

52- 8917 

53- 

53- 0200 



5174 
6469 

7759 
9045 



5304 
6598 

7888 
9174 



5434 
6727 

8016 

9302 



5563 
6856 

8145 
9430 



5693 
6985 

8274 
9559 



5822 
7114 
8402 
9687 



5951 

7243 

8531 
9815 



6081 

7372 
8660 

9943 



0328 0456 0584 0712 



0840 0968 1096 1223 



6210 

7501 
8788 

0072 
1351 



[29 
[29 

129 
[28 
[28 
28 



840 

341 
342 
343 
344 



53- 1479 
53- 2754 
53- 4026 

53- 5294 
53- 6558 



1607 
2882 

4153 
5421 
6685 



1734 
3009 

4280 

5547 
6811 



1862 

3136 
4407 

5674 
6937 



1990 
3264 

4534 
5800 

7063 



2117 

3391 
4661 

5927 
7189 



2245 
3518 
4787 
6053 
7315 



2372 

3645 
4914 

6180 
7441 



2500 

3772 
5041 
6306 

7567 



2627 

3899 
5167 

6432 
7693 



[28 

[27 

[27 

[26 

[26 



346 

346 
346 

347 
348 

349 



53- 7819 

53- 9076 

54- 

54- 0329 

54- 1579 
54- 2825 



7945 
9202 

0455 
1704 

2950 



8071 
9327 

0580 
1829 

3074 



8197 
9452 

0705 

1953 
3199 



8322 
9578 

0830 
2078 

3323 



8448 
9703 

•  • • • 

0955 
2203 

3447 



8574 
9829 

1080 
2327 

3571 



8699 
9954 

1205 

2452 
3696 



8825 8951 



0079 

1330 
2^76 
3820 



0204 

1454 
2701 

3944 



[26 
[26 
[25 

[25 

25 
124 



360 

351 
352 
353 
354 
354 



365 

356 
357 

N 



54- 4068 

54- 5307 
54- 6543 

54- 7775 

54- 9003 

55- 



4192 

5431 
6666 

7898 

9126 



4316 

5555 
6789 
8021 

9249 



4440 
5678 
6913 
8144 

9371 



4564 
5802 

7036 

8267 

9494 



4688 

5925 

7159 

8389 
9616 



4812 
6049 
7282 
8512 

9739 



4936 
6172 

7405 

863s 
9861 



5060 
6296 

7529 

8758 
9984 



5183 
6419 
7652 
8881 

0106 



55- 0228 0351 0473 0595 0717 
55- '450 1572" 1694 1816 1938 
55- 2668 2790 29 I I 3033 3155 



0840 
2060 
3276 



0962 
2181 
3398 



1084 
2303 

3519 



1206 

2425 
3640 

8 



1328 

2547 
3762 



[24 

124 
'23 
23 
23 
'23 



[22 

[22 
[21 

D 



LOGARITHMS OF NUMBERS. 



45 



N 


I 


2 


3 


4 


5 


6 


7 


8 9 


D 
121 


358 


55- 3883 4004 


4126 


4247 


4368 


4489 


4610 


4731 


4852 4973 


359 


55- 5094 52 » 5 


5336 


5457 


5578 


5699 


5820 


5940 


6061 6182 


121 


360 55- 6303 6423 


6544 


6664 


6785 


6905 
8108 


7026 


7146 


7267 7387 


120 


361 55- 7507 7627 


7748 


7868 


7988 


8228 


8349 


8469 8589 


120 


362 1 55- 8709 8829 


8948 


go68 


91^ 


9308 


9428 


9548 


9667 9787 


120 


363 55- 9907 
















120 


363 


56- 0026 


0146 


0265 


0385 


0504 
1698 


0624 


0743 


0863 0982 


"9 


364 


56- iioi 1221 


1340 


H59 


1578 


1817 


1936 


2055 2174 


119 


365 56- 2293 2412 


2531 


2650 


2769 


2887 


3006 


3125 


3244 3362 


119 


366 56- 3481 3600 


37»8 


3837 


3955 


4074 


4192 


43" 


4429 4548 


119 


367 


56- 4666 4784 


4903 


5021 


5139 


5257 


5376 


5494 


5612 5730 


118 


368 


56- 5848 5966 


6084 


6202 


6320 


6437 


6555 


6673 


6791 6909 


118 


369 


56- 7026 7144 


7262 


7379 


7497 


7614 


7732 


7849 


7967 8084 


118 


370 56- 8202 8319 


8436 


8554 


8671 


8788 


8905 


9023 


9140 9257 


117 


371 


55- 9374 9491 


9608 


9725 


9842 


9959 




^ 




117 


37 « 


57- -]•.. 










0076 


0193 


0309 0426 


"7 


372 


57- 0543 0660 


0776 


0893 
2058 


lOIO 


1 126 


1243 


1359 


1476 1592 


117 


373 


57- 1709 1825 


1942 


2174 


2291 


2407 


2523 


2639 2755 


116 


374 57- 2872 2988 


3»04 


3220 


3336 


3452 


3568 


3684 


3800 3915 


116 


375 57- 403' 4147 


4263 


4379 


4494 


4610 


4726 


4841 


4957 5072 


116 


376 


57- 5188 5303 


S419 


^^ 


5650 


5765 


5880 


5996 


61 1 1 6226 


"5 


377 


57- 6341 6457 


6572 


6802 


6917 


7032 


7147 
8295 


7262 7377 


"5 


378 


57- 7492 7607 


7722 


7836 


7951 


8066 


81S1 


8410 8525 


"5 


379 


57- 8639 8754 


8868 


8983 


9097 


9212 


9326 


9441 


9555 9669 


114 


380 57- 9784 9898 














■• «■■ •«■••■ 


"4 


380 58- 


0012 


0126 


0241 


0355 


0469 


0583 


0697 081 I 


114 


381 


58- 092s 1039 


"53 


1267 


1381 


1495 


1608 


1722 


1836 1950 
2972 3085 


"4 


382 


58- 2063 2177 


2291 


2404 


2518 


2631 


2745 


2858 


114 


383 


58- 3199 33»2 


3426 


3539 


3652 


3Z^§ 


3879 


3992 


4105 4218 


"3 


^ 


58- 4331 444* 


4557 


4670 


4783 


4896 


5009 


5122 


5235 5348 


"3 


385 


58- 5461 5574 


5686 


5799 


5912 


6024 


6137 


6250 


6362 6475 


"3 


386 


58- 6587 6700 


6812 


6925 


7037 


7149 


P^^ 


7374 


7486 7599 


112 


387 


58- 77" 7823 


7935 


8047 


8160 


8272 


8384 


8496 


8608 8720 


112 


38S 


58- 8832 8944 


9056 


9167 


9279 


9391 


9503 


9615 


9726 9838 


112 


. 389 


58- 9950 
















112 


389 


59- 0061 


0173 


0284 


0396 


0507 


0619 


0730 


0842 0953 


112 


300 


59- 1065 1 1 76 


1287 


1399 


15 10 


I62I 


1732 


1843 


1955 2066 


III 


391 


59- 2177 2288 


2399 


2510 


2621 


2732 


2843 


2954 


3064 3175 


III 


392 


59- 3286 3397 


3«>o8 


3618 


3729 


3840 


3950 


4061 


41 7 I 4282 


III 


393 


59- 4393 4503 


4614 


4724 


4834 


4945 


5055 


5165 


5276 5386 


110 


394 


59- 5496 . 5606 


5717 


5827 


5937 


6047 


6157 


6267 


6377 6487 


no 


395 


59- 6597 6707 


6817 


6927 


7037 


7146 


7256 


7366 


7476 7586 


no 


396 


59- 7695 7805 
59- 879* 8900 


7914 


8024 


8134 


8243 


8353 


8462 


8572 8681 


no 


397 


9009 


9119 


9228 


9337 


9446 


9556 


9665 9774 


109 


398 


59- 9883 9992 
















109 


398 


60- 


OIOI 


0210 


0319 I 0428 


0537 


0646 


0755 ^864 


109 


399 
400 


60- 0973 1082 


II9I 


1299 


1408 ^ 1517 


1625 


1734 


1843 1951 


109 


60- 2060 2169 


2277 


2386 


2494 


2603 

5 


271 1 
6 


2819 

7 


2928 3036 
8 9 


108 


N 

1 


I 


2 


3 


4 



46 



MATHEMATICAL TABLES. 



N 





I 


2 


3 


4 


5 


6 


7 


8 


9 


D 

108 


401 


60- 3144 


3253 


3361 


3469 


3577 


3686 


3794 


3902 


4010 


41 18 


402 


60- 4226 


4334 


4442 


^§50 


4658 


4766 


4874 


4982 


5089 


5197 


108 


403 


60- 5305 


5413 


5521 


5628 


5736 


5844 


595' 


6059 


6166 


6274 


108 


404 


60- 6381 


6489 


6596 


6704 


6811 


6919 


7026 


7133 


7241 


7348 


107 


405 


60- 7455 


7562 


7669 


SJ? 


7884 


\ 7991 


8098 


8205 


8312 


8419 


107 


406 


60- 8526 


8633 


8740 


8954 ; 9061 


9167 


9274 


9381 


9488 


107 


407 


60- 9594 


9701 


9808 


9914 














107 


407 


61- 








0021 


0128 


0234 


0341 


0447 


0554 


107 


408 


6x- 0660 


0767 


0^73 


0979 


1086 


1 192 


1298 


1405 


1511 


1617 


106 


409 


61- 1723 


1829 


1936 


2042 


2148 


2254 


2360 


2466 


2572 


2678 


X06 


410 


61- 2784 


2890 


2996 


3102 


3207 


3313 


3419 


3525 


3630 
4686 


3736 


106 


411 


61- 3842 


3947 


4053 


4159 


4264 


4370 


4475 


4581 


4792 


106 


412 


61 - 4897 


5003 


5108 


5213 


5319 


5424 


5529 


5634 


5740 


5845 


105 


413 


61- 5950 


6055 


6160 


6265 


6370 


6476 


6581 


6686 


6790 


6895 


105 


414 


61- 7000 


7105 


7210 


7315 


7420 


7525 


7629 


7734 


7839 


7943 


105 


415 


61- 8048 


8153 
9198 


8257 


8362 


8466 


8571 


8676 


8780 


8884 


8989 


105 


416 


61-9093 


9302 


9406 


9511 


9615 


9719 


9824 


9928 




105 


4x6 


6a- 












«■•••• 






0032 


104 


417 


62- 0136 


0240 
1280 


0344 


0448 


0552 


0656 


0760 


0864 


0968 


1072 


104 


418 


62- 1 1 76 


1384 


1488 


1592 


1695 


1799 


1903 


2007 


21 10 


104 


419 


62- 2214 


2318 


2421 


2525 


2628 


2732 


2835 


2939 


3042 


3146 


104 


420 


62- 3249 


3353 


4488 


3559 


3663 


3766 


3869 


3973 


4076 


4179 


103 


421 


62- 4282 


4385 


4591 


4695 


4798 


4901 


5004 


5107 


5210 


103 


422 


6a- 5312 


5415 


5518 


5621 


5724 


5827 


5929 


6032 


6135 


6238 


»03 


423 


62- 6340 


6443 


6546 


6648 


6751 


^^53 


6956 
7980 


Z°5^ 


7161 


P^^ 


103 


424 


62- 7366 


7468 


7571 


7673 


7775 


7878 


8082 


8185 


8287 


102 


425 


62- 8389 


8491 


8593 


8695 


8797 


8900 


9002 


9104 


9206 


9308 


102 


426 


62- 9410 


9512 


9613 


9715 


9817 


9919 










102 


426 


63- 












0021 


0123 


0224 


0326 


102 


427 


63- 0428 


0530 


0631 


0733 


^35 


0936 


1038 


1 139 


1241 


1342 


102 


428 


63- 1444 


1545 


1647 


1748 


1849 


1951 


2052 


2153 


2255 


2356 


lOI 


429 


63- 2457 


2559 


2660 


2761 


2862 


2963 


3064 


3«65 


3266 


3367 


lOI 


480 


63- 3468 


3569 


3670 


3771 


3872 


3973 


4074 


4175 


4276 


4376 


lOI 


431 


63- 4477 


4578 


4679 


4779 


4880 


4981 


5081 


5182 


5283 


5383 


lOI 


432 


63- 5484 


5584 


5685 


5785 


5886 


5986 


6087 


6ih7 


6287 


6388 


100 


433 


63- 6488 


6588 


6688 


6789 


6889 


6989 


7089 


7189 


7290 


7390 
8389 


100 


434 


63- 7490 


7590 


7690 


7790 


7890 


7990 


8090 


8190 


8290 


100 


485 


63- 8489 


8589 


8689 


8789 


8888 


8988 


9088 


9188 


9287 


9387 


100 


436 


63- 9486 


9586 


9686 


9785 


9885 


9984 










lOO 


436 


64- 












0084 


0183 


0283 


0382 


99 


437 


64- 0481 


0581 


0680 


0779 


0879 


0978 


1077 


"77 


1276 


1375 


99 


438 


64- 1474 


1573 


1672 


1771 


1871 


1970 


2069 


2168 


2267 


2366 


99 


439 


64- 2465 


2563 


2662 


2761 


2860 


2959 
3946 


3058 
4044 


3156 
4143 


3255 
4242 


3354 
4340 


99 
99 


440 


64- 3453 


3551 


3650 


3749 


3847 


441 


64- 4439 


4537 


4636 


4734 


4832 


4931 


5029 


5127 


5226 


5324 


98 


442 


64- 5422 


5521 


5619 


5717 


5815 


5913 


601 1 


6110 


6208 


6306 


98 


443 


64- 6404 


6502 


6600 


6698 


6796 


6894 


6992 


7089 


7187 


7285 


98 


444 

N 


64- 7383 


7481 


7579 


7676 


7774 


7872 


7969 


8067 


8165 


8262 


98 





I 


2 


3 


4 


s 


6 


7 


8 


9 


D 



LOGARITHMS OF NUMBERS. 



47 



N o 

P 


I 


2 


3 


4 


S 


6 


7 


8 


9 


D 


445 


64- 8360 


8458 


8555 


8653 


8750 


8848 


8945 


9043 


9140 


9237 


97 


446 


64- 9335 


9432 


9530 


9627 


9724 


9S21 


9919 








97 


446 


65- 














0016 


0113 


0210 


97 


447 


65- 0308 


0405 


0502 


0599 


0696 


0793 


0890 


0987 


1084 


1181 


97 


i 44S 


65- 1278 


1375 


1472 


1569 


1666 


1762 


'S^? 


1956 


2053 


2150 


97 


449 


65- 2246 


2343 


2440 


2536 


2633 


2730 


2826 


2923 


3019 


311^ 


97 


450 


65- 3213 


3309 


3405 


3502 


3598 


3695 


3791 


3888 


3984 


4080 


96 


4S» 


65- 4177 


4273 


4369 


4465 


4562 


4658 


4754 


4850 


4946 


5042 


96 


452 


65- 5138 


5235 


5331 


5427 


5523 


5619 


5715 


5810 


5906 


6002 


96 


453 


65- 6098 


6194 


6290 


6386 


6482 


6577 


6673 


6769 


6864 


6960 


96 


454 65- 7056 


7152 


7247 


7343 


7438 


7534 


7629 


7725 


7820 


7916 


96 


455 65- Soil 


8107 


8202 


8298 


8393 


8488 


8584 


8679 


8774 


8870 


95 


456 


65- 8965 


9060 


9155 


9250 


9346 


9441 


9536 


963^ 


9726 


9821 


95 


457 


65- 9916 










••■•»■ 










95 


457 


66- 


OOII 


0106 


0201 


0296 


0391 


0486 


0581 


0676 


0771 


95 


45^ 


66- 0865 


0960 


1055 


1 150 


1245 


1339 


1434 
2380 


1529 


1623 


1718 


95 


459 


66- 1813 


1907 


2002 


2096 


2191 


2286 


2475 


2569 


2663 


95 


460 


65- 2758 


2852 


2947 


3041 


3135 


3230 


3324 


3418 


3512 


3607 


94 


461 


66- 3701 


3795 


3889 


3983 


4078 


4172 


4266 


4360 


4454 


4548 


94 


462 


66- 4642 


4736 


4830 


4924 


5018 


5112 


5206 


5299 


5393 


5487 


94 


463 


66- 5581 


5675 


5769 


5862 


5956 


6050 


6143 


6237 


6331 


6424 


94 


464 


66- 6518 


6612 


6705 


6799 


6892 


6986 


7079 


7173 


7266 


7360 


94 


465 


66- 7453 


7546 


7640 


7733 


7826 


7920 


8013 


8106 


8199 


8293 


93 


466 


66- 8386 


8479 


8572 


8665 


8759 


88^2 


8945 


9038 


9J31 


9224 


93 


467 


66- 9317 


9410 


9503 


9596 


9689 


9782 


9875 


9967 






93 


467 


67- 






• • • a • 










0060 


0153 
1080 


93 


46S 


67- 0246 


0339 


0431 


0524 


0617 


0710 


0802 


089s 


0988 


93 


469 


67- "73 


1265 


1358 


145 I 


1543 


1636 


1728 


1821 


1913 


2005 


93 


470 


67- 2098 


2190 


2283 


2375 


2467 


2560 


2652 


2744 


2836 


2929 


92 


471 


67- 3021 


3"3 


3205 


3297 


3390 


3482 


3574 


3666 


3758 


3850 


92 


472 


67- 3942 


4034 


4126 


4218 


4310 


4402 


4494 


4586 


4677 


4769 


92 


473 


67- 4861 


4953 


5045 


5137 


5228 


5320 


5412 


5503 


5595 


5687 


92 


474 


67- 5778 


5870 


5962 


6053 


6145 


6236 


6328 


6419 


6511 


6602 


92 


475 67- 6694 


6785 
7698 


6876 


6968 


7059 


7151 


7242 
8154 


7333 


7424 
8336 


7516 


91 


476 


67- 7607 
67- 8518 


7789 


7881 


7972 


8063 


8245 


8427 


91 


 477 


8609 


8700 


8791 


8882 


^Z3 


9064 


9155 


9246 


9337 


91 


478 


67- 9428 


9519 


9610 


9700 


9791 


9882 


9973 








91 


478 


68- 














C063 


0154 


0245 


91 


479 


68- 0336 


0426 


0517 


0607 


0698 


0789 


0879 


0970 


1060 


1151 


91 


480 


68- 1241 


1332 


1422 


I5I3 
2416 


1603 


1693 


1784 


1874 


1964 


2055 


90 


481 


68- 2145 


2235 


2326 


2506 


2596 


2686 


2777 


2867 


2957 


90 


4S2 


68- 3047 


3137 


3227 


3317 


3407 


3497 


3557 


3^77 


3767 


3857 


90 


: 483 68- 3947 


4037 


4127 


4217 


4307 


4396 


4486 


4576 


4666 


4756 


90 


484 


68- 4845 


4935 


5025 


5II4 


5204 


5294 


5383 


5473 


5563 


5652 


90 


485 


68- 5742 


5831 


5921 


6010 


61CX) 


6189 


6279 


6368 


6458 


6547 


89 


486 


68- 6636 


6726 


6815 


6904 


6994 


7083 


7172 


7261 


7351 


7440 


89 


487 


68- 7529 


7618 


7707 


7796 


7886 


7975 


8064 


8153 


8242 


8331 


89 


488 


68- S420 


8509 


8598 


8687 


8776 


8865 


8953 


9042 


9131 


9220 


89 


1489 


68- 9309 


9398 


9486 


9575 


9664 


9753 
5 


9841 


9930 






89 
D 


j N } 


I 


2 


3 


4 


6 


7 


8 


9 



48 



MATHEMATICAL TABLES. 



N 

4«9 

1490 

491 
492 

493 
494 



69- 



8 



0019 0107 



69- 0196 
69- 1081 
69- 1965 
69- 2847 

69-3727 



0285 
1 170 

2053 
2935 
381S 



0373 
1258 

2142 

3023 

3903 



0462 

1347 
2230 

3111 
3991 



0550 

1435 
2318 

3>99 
4078 



0639 

1524 
2406 

3287 
4166 



0728 
1612 
2494 

3375 
4254 



0816 
1700 

2583 
3463 
4342 



0905 

17S9 
2671 

3551 
4430 



0993 
1877 

2759 
3639 
4517 



D 



89 



89 
88 

88 

88 

88 



496 

496 

497 
498 
499 



69- 4605 
69- 5482 
69- 6356 
69- 7229 
69- 8101 



4693 
5569 
6444 

7317 
8188 



4781 
5657 

6531 
7404 

8275 



4868 

5744 
6618 

7491 
8362 



4956 
5832 
6706 

7578 
8449 



5044 

5919 

6793 
7665 

8535 



513" 
6007 

6880 

7752 
8622 



5219 
6094 
6968 

7839 
8709 



5307 
6182 

7055 
7926 

8796 



5394 
6269 

7142 

8014 

8883 



88 

87 
87 
87 
87 



500 

501 

502 

503 
504 



69- 8970 

69- 9838 

70- 

70- 0704 
70- 1568 
70- 2431 



9057 
9924 

0790 
1654 
2517 



9144 9231 9317 



001 1 
0877 
1741 
2603 



0098 

0963 
1827 

2689 



0184 
1650 

1913 
2775 



9404 9491 9578 9664 9751 



0271 
1 136 
1999 
2861 



0358 
1222 

2086 

2947 



0444 

1309 
2172 

3033 



0531 

1395 
2258 

3"9 



0617 
1482 

2344 
3205 



87 

87 

87 
86 

86 

86 



505 

506 

507 
508 

509 



510 

511 
512 

512 

5*3 
514 



70- 3291 

70- 4151 
70- 5008 
70- 5864 
70- 6718 



3377 
4236 

5094 

5949 
6803 



3463 
4322 

5179 
60- 

6^ 



3549 
4408 

5265 
6120 

6974 



3635 

4494 

5350 
6206 

7059 



3721 
4579 
5436 
6291 

7144 



3807 
4665 
5522 
6376 
7229 



3893 

4751 
5607 

6462 
7315 



3979 

4837 

5693 

6547 
7400 



4065 

4922 

5778 
6632 

7485 



70- 7570 
70- 8421 

70- 9270 

71- 

71- 0117 

71- 0963 



7655 
8506 

9355 

0202 
1048 



7740 

8591 
9440 

0287 
1 132 



7826 
8676 

9524 

0371 
1217 



791 1 
8761 
9609 

0456 
1301 



7996 
8846 
9694 

0540 
1385 



8081 

8931 
9779 

0625 
1470 



8166 

9015 
9S63 

0710 
1554 



8251 
9100 
9948 

0794 
1639 



8356 
9185 

• • • •  • 

0033 
0879 

1723 



86 
86 
86 

85 
85 



f5 

85 
85 

84 



516 
516 

518 
5«9 



71- 1807 
71- 2650 

71- 3491 
71- 4330 
71- 5*67 



1892 

2734 

3575 
4414 

5251 



1976 
2818 

3659 
4497 
5335 



2060 
2902 

3742 
4581 
5418 



2144 
2986 
3826 
4665 

5502 



2229 
3070 

3910 
4749 
5586 



2313 

3>54 

3994 

4833 
5669 



2397 
3238 

4078 

4916 

5753 



2481 

3323 
4162 

5000 
5836 



2566 

3407 
4246 

5084 
5920 



84 
84 
84 
84 
84 



|520 

, 521 
522 

i 523 
524 
524 



71- 6003 
71- 6838 
71- 7671 
71- 8502 

71- 9331 

72- 



6087 
6921 

7754 

8585 
9414 



6170 
7004 

7837 
8668 

9497 



6254 
7088 
7920 
8751 
9580 



6337 
7171 
8003 

8834 
9663 



6421 

7254 
8086 

8917 
9745 



6s(H 

7338 
8169 
9000 
9828 



6588 

7421 
8253 

9083 
9911 



6671 

7504 
8336 
9165 

9994 



6754 

7587 
8419 

9248 
0077 



7a- 01 g9 0242 0325 0407 0490 

72- 0986 1068 I 151 1233 1316 

7a- i8u 1893 1975 2058 2140 

7a- 2634 2716 2798 2881 2963 

72- 3456 3538 3620 3702 3784 



0573 0655 0738 0821 0903 

1398 1481 1563 1646 1728 

2222 2305 2387 2469 2552 

3045 3127 3209 3291 3374 

3866 3948 4030 41 12 4194 



72- 4276 


4358 


4440 


4522 


4604 


72- 5095 


5176 


5258 


5340 


5422 


72- 5912 


5993 


6075 


61 s6 


6238 


7a- 6727 


0^ 


6890 


6972 


7053 



4685 4767 4849 4931 50'3 

5503 5585 5667 5748 5830 

6320 6401 64S3 6564 6646 

7134 7216 7297 7379 7460 



«3 

83 

P 
83 



83 

82 

82 
82 
82 



82 
82 
82 
81 



8 



D 



LOGARITHMS OF NUMBERS. 



49 



1 

1 

1« 


N 

534 





I 


2 


3 


4 


5 


6 


7 


8 


9 


D 
81 
81 


72- 7541 


7623 


7704 


7785 


7866 


7948 


8029 


8110 


8191 


8273 
9084 


»d5 1 


7^ 8354 


8435 


8516 


8597 


8678 


8759 


8841 


8922 


9003 


536 


72- 9165 


9246 


9327 


9408 


9489 


9570 


9651 


9732 


9813 


9893 


81 


537 


72- 9974 










^ 










81 


537 73- 


OOS5 


0136 


0217 


0298 


0378 


0459 


0540 


0621 


0702 


81 


1 538 1 


73- 0782 


086^ 


0944 


1024 


1105 


1186 


1266 


1347 


1428 


1508 


81 


.{ 


539 


73- 1589 


1669 


1750 


1830 


1911 


1991 


2072 


2152 


2233 


2313 


81 


HO 


73- 2394 


2474 


2S55 


2635 
3438 


2715 


2796 


2876 


2956 


3037 


3"7 


80 




541 


73- 3>97 


3278 


3358 


3518 


3598 


3679 


3759 


3839 


3919 


80 




542 


73- 3999 


4079 


4160 


4240 


4320 


4400 


4480 


4560 


4640 


4720 


80 




543 


73- 4800 


4880 


4960 


5040 


5120 


5200 


5279 


5359 


5439 


55>9 


80 


I 


544 


73- 5599 


5679 


5759 


5838 


5918 


5998 


6078 


6157 


6237 


6317 


io 


546 


73- 6397 


6476 


6556 


6635 


6715 


6795 


6874 


6954 


7034 


7113 


80 




546 


73- 7193 


7272 


7352 


7431 


751 1 


7590 


7670 


7749 


7829 


7908 


79 




547 


73- 7987 


8067 


8x46 


8225 


8305 


8384 


8463 


8543 


8622 


8701 


79 




548 


73- 8781 


8860 


8939 


9018 


9^7 


9'?Z 


9256 


9335 


9414 


9493 


79 




549 


73- 9572 


9651 


9731 


9810 


9889 


9968 










79 




549 


74- 












0047 


0126 


0205 


0284 


79 


i 


650 


74- 0363 


0442 


0521 


0600 


0678 


0757 


0836 


0915 


0994 


'2?3 


79 


.'551 


74- 1152 


1230 


1309 


1388 


1467 


1546 


1624 


1703 


1782 


i860 


79 




552 


74- 1939 


2018 


2096 


2175 


2254 


2332 


241 1 


2489 


2568 


2647 


79 




553 74- 2725 


2804 


2882 


2961 


3039 


3II8 


3196 


3275 


3353 


3431 


78 




554 


74- 35>o 


3588 


3667 


3745 


3823 


3902 


3980 


4058 


4136 


4215 


78 


n 


555 1 74- 4293 


4371 


4449 


4528 


4606 


4684 


4762 


4840 


4919 


4997 


75 




556 74- 5075 


5153 


5231 


5309 


5387 


S465 


5543 


5621 


5699 


5777 


78 




557 


74- 5855 


5933 


601 1 


6089 


6167 


6245 


6323 


6401 


6479 


6556 


78 




558 


74- 6634 


6712 


6790 


6868 


6945 


7023 


7101 


7179 


7256 


7334 


78 




559 


74- 7412 


7489 


7567 


7645 


7722 


7800 


7878 


7955 


8033 


8110 


78 


560 


74 81S8 


8266 


8343 


8421 


8498 


8576 


8653 


8731 


8808 


8885 


77 


;50i 


74-8963 


9040 


9118 


9195 


9272 


9350 


9427 


9504 


9582 


9659 


77 


562 i 74- 9736 


Q814 


9891 














77 


562 ; 75- 








0045 


0123 


0200 


0277 


0354 


0431 


77 N 


563 


75- 0508 


0586 


ii663 


0740 


0817 


0894 


0971 


1048 


1 125 


1202 


77 




■564 


75- 1279 


1356 


1433 


1510 


1587 


1664 


1 741 


1818 


1895 


1972 


77 




566 


75- 2048 


2125 


2202 


2279 


2356 


2433 


2509 


2586 


2663 


2740 


77 




>66 


75- 2816 


2893 


2970 


3047 


3123 


3200 


3277 


3353 


3430 


3506 


77 




'^ 


75- 3583 


S66o 


3736 


3813 


3889 


3966 


4042 


41 19 


4195 


4272 


77 




568 


75- 4348 


4425 


4501 


4578 


4654 


4730 


4807 


4883 


4960 


5036 


76 




S^ 


75- 5"2 


5189 


5265 


5341 


5417 


5494 


5570 


5646 


5722 


5799 


76 




570 


75- 5875 


595 « 


6027 


6103 


6180 


6256 


6332 


6408 


6484 


6560 


76 




571 


75- 6636 


6712 


6788 


6864 


6940 


7016 


7092 


7168 


7244 


7320 
227? 


76 




572 


75- 7396 


7472 


7548 


7624 


7700 


7775 


7851 


7927 


8003 


76 




573 ' 75- 8155 


8230 
8988 


8306 


8382 


8458 


8533 


8609 


8685 


8761 


8836 


76 




574 , 75- 8912 


9063 


9139 


9214 


9290 


9366 


9441 


9517 


9592 


76 


575 


75- 9668 


9743 


9819 


9894 


9970 












76 


575 


75- 








•  » • • 


0045 


0121 


0196 


0272 


0347 


75 


576 


75- 0422 


0498 


0573 


0649 


0724 


0799 


0875 


0950 


1025 


IIOI 


75 




577 

N 


76- 1176 


1251 


1326 


1402 


1477 


1552 


1627 


1702 


1778 


1853 


75 
D 







I 


2 


3 


4 


5 


6 


7 


8 


9 



so 



MATHEMATICAL TABLES. 



N 
578 





I 


2 


3 


4 


5 


6 


7 


8 


9 


D 

75 


76- 1928 


2003 


2078 


2153 


2228 


2303 


2378 


2453 


2529 


2604 


579 


76- 2679 


2754 


2829 


2904 


2978 


3053 

• 


3128 


3203 


3278 


3353 


75 


580 


76- 3428 


3503 


3578 


3653 


3727 


3802 


3877 


3952 


4027 


4101 


75 


5^' 


76- 4176 


4251 


4326 


4400 


4475 


4550 


4624 


4699 


4774 


4848 


75 


5^^ 


76- 4923 


4998 


5072 


5147 


5221 


5296 


5370 


5445 


5520 


5594 


75 


553 


76- 5669 


5743 


5818 


5892 


5966 


6041 


6115 


6190 


6264 


6338 
7082 


74 


584 


76- 6413 


6487 


6562 


6636 


6710 


6785 


6859 


6933 


7007 


74 
74 


585 


76- 7156 


7230 


7304 


7379 


7453 


7527 


7601 


7675 


7749 


7823 


586 


76- 7898 


7972 


' 8046 


8120 


8194 


8268 


8342 


8416 


8490 


8564 


74 


5^Z 


76- 8638 


8712 


8786 


8860 


8934 


9008 


9082 


9156 


9230 


9303 


74 


588 


76- 9377 


9451 


9525 


9599 


9673 


9746 


9820 


9894 


9968 




74 


588 


77- 






•••••• 












0042 


74 


589 


77- 0115 


0189 


0263 


0336 


0410 


0484 


0557 


0631 


0705 


0778 


74 


590 


77- 0852 


0926 


0999 


IP73 


1 146 


1220 


129^ 


1367 


1440 


1514 


74 


591 


77- 1587 


1661 


1734 


180B 


1881 


^9?S 


2028 


2102 


2175 


2248 


73 


592 


77- 2322 


2395 


2468 


2542 


2615 


2688 


2762 


2835 


2908 


2981 


73 


593 


77- 3055 


3128 


3201 


3274 


3348 


3421 


3494 


3567 
4298 


3640 


3713 


73 


594 


77- 3786 


3860 


3933 


4006 


4079 


4152 


4225 


4371 


4444 


73 


595 


77- 4517 


4590 


4663 


4736 


4809 


48S2 


4955 


5028 


5100 


5173 


73 


596 


77- 5246 


5319 


5392 


5465 


5538 


5610 


5683 


5756 


5829 


5902 


73 


597 


77- 5974 


6047 


6120 


6193 


6265 


6338 


641 1 


6483 


6556 


6629 


73 


598 


77- 6701 


6774 


6846 


6919 


6992 


7064 


7137 


7209 


7282 


7354 


73 


599 


77- 7427 


7499 


7572 


7644 


77'7 


7789 


7862 


7934 


8006 


8079 


72 


600 


77- 8151 


i^224 


8296 


8368 


8441 


8513 


8585 


8658 
9380 


8730 


8802 


72 


601 


77- 8874 


8947 


9019 


9091 


9163 


9236 


9308 


9452 


9524 


72 


602 


77- 9596 


9669 


9741 


9813 


9885 


9957 








• • • • « • 


72 


602 


78- 












0029 


OIOI 


0173 


0245 


72 


603 


78- 0317 


0389 


0461 


0533 


0605 


0677 


0749 


0821 


0893 


0965 


72 


604 


78- 1037 


IIO9 


1181 


1253 


1324 


1396 


1468 


1540 


1612 


1684 


72 


605 


78- 1755 


1827 


1899 


1971 


2042 


2114 


2186 


2258 


2329 


2401 


72 


606 


78- 2473 


2544 


2616 


2688 


2759 


2831 


2902 


2974 


3046 


3117 


72 


607 


78- 3189 


3260 


3332 


340.S 


3475 


3546 


3618 


3689 


3761 


3832 


71 


608 


78- 3904 


3975 


4046 


41 18 


4189 


4261 


4332 


4403 


4475 


4546 


71 


609 


78- 4617 


4689 


4760 


4831 


4902 


4974 


5045 


5116 


5187 


5259 


71 


610 


78- 5330 


5401 


5472 


5543 


5615 


5686 


5757 


5828 


5899 


5970 


71 


611 


78- 6041 


6112 


6183 


6254 


6325 


6396 


6467 


6538 


6609 


6680 


71 


612 


78- 6751 


6822 


6893 


6964 


7035 


7106 


7177 


7248 


7319 


7390 


71 


613 


78- 7460 


753> 


7602 


7673 


7744 


7815 


7885 


7956 


8027 


8098 


71 


614 


78- 8168 


8239 


8310 


8381 


84s I 


8522 


8593 


8663 


8734 


8804 


71 


615 


78- 8875 


8946 


9016 


9087 


9157 


9228 


9299 


9369 


9440 


9510 


71 


616 


78- 9581 


9651 


9722 


9792 


9863 


9933 










70 


tfi6 


79- 












cxx}4 


0074 


0144 


0215 


70 


617 


79- 0285 


0356 


0426 


0496 


0567 


0637 


0707 


0778 


0848 


0918 


70 


618 


79- 0988 


1059 


1 129 


1 199 


1269 


1340 


1410 


1480 


1550 


1620 


70 


619 
620 


79- I 69 I 


1761 


1831 


1901 


1971 


2041 


2111 


2181 


2252 


2322 


70 


79- 2392 


2462 


2532 


2602 


2672 


2742 


2812 


28S2 


2952 


3022 


70 


621 


79- 3092 


3162 


3231 


3301 


3371 


3441 


35" 


3581 


3651 


3721 


70 


622 

N 


79- 3790 


3860 


3930 


4000 


4070 


4139 


4209 


4279 


4349 


4418 


70 
D 





I 


2 


3 


4 


_ 5 


6 


7 


8 


9 



LOGARITHMS OF NUMBERS. 



SI 



Nj 





I 

« 


2 


3 


4 


5 


6 


7 


'8 


9 


D 

70 


623 


79- 4488 


4558 


4627 


4697 


4767 


4836 


4906 


4976 


5045 


5"S 


624 79- 5185 


5254 


5324 


5393 


5463 


5532 


5602 


5672 


5741 


5811 


70 


625 79- 5880 


5949 


6019 


6088 


6158 


6227 


6297 


6366 


6436 


6505 


69 


626 79- 6574 


6644 


6713 


6782 


6852 


6921 


6990 


7060 


7129 


7198 


69 


627 


79- 7268 


7337 


7406 


7475 


7545 


7614 


7683 


7752 


7821 


7890 


69 


628 


79- 7960 


8029 


8098 


8167 


8236 


8305 


8374 


8443 


8513 


8582 


69 


629 


79-8651 


8720 


8789 


8858 


8927 


8996 


9065 


9134 


9203 


9272 


69 


630 

631 


79- 9341 


9409 


9478 


9547 


9616 


9685 


9754 


9823 


9892 


9961 


69 


80- 0029 


0098 


0167 


0236 


0305 


0373 


0442 


0511 


0580 


0648 


69 


632 


80- 0717 


0786 


o8s4 


0923 


0992 


1061 


1 129 


X198 


1266 


1335 


69 


633 


80- 1404 


1472 


1541 


1609 


1678 


1747 


1815 


1884 


1952 


2021 


69 


634 


80- 2089 


2158 


2226 


2295 


2363 


2432 


2500 


2568 


2637 


2705 


69 


635 


80-2774 


2842 


2910 


2979 


3047 


3116 


3184 


3252 


3321 


3389 


68 


636 


80-3457 


3525 


3594 


3662 


3730 


3798 


3867 


3935 


4003 


4071 


68 


637 


80- 4139 


4276 


4344 


4412 


4480 


4548 


4616 


4685 


4753 


68 


638 


8a- 4821 


4889 


4957 


5025 


5093 


5161 


5229 


5297 


5365 


5433 


68 


639 


80- 5501 


5569 


5637 


5705 


5773 


5841 


5908 


5976 


6044 


6112 


68 


640 


80- 6180 


6248 


6316 


6384 


6451 


6519 


6587 


6655 


6723 


6790 


68 


641 


80- 6858 


6926 


6994 


7061 


7129 


7197 


7264 


7332 


7400 


7467 


68 


642 


80- 7535 


7603 


7670 


7738 


7806 


7873 


7941 


8008 


8076 


8143 


68 


643 


80- 821 1 


8279 


8346 


8414 


8481 


8549 


8616 


8684 


8751 


8818 


67 


644 


80- 8886 


8953 


9021 


9088 


9156 


9223 


9290 


9358 


9425 


9492 


67 


645 


80- 9560 


9627 


9694 


9762 


9829 


9896 


9964 


• • • • • 






67 


^l 


81- 














0031 


0098 


0165 


67 


646 


81- 0233 


0300 


0367 


0434 


0501 


0569 


0636 


0703 


0770 


0837 


67 


^7 


81- 0904 


0971 


1039 


1 106 


"73 


1240 


1307 


1374 


144 1 


1508 


67 


648 


81- 1575 


1642 


1709 


1776 


1843 


1910 


1977 


2044 


2111 


2178 


67 


649 1 8x- 2245 


2312 


2379 


2445 


2512 


2579 


2646 


2713 


2780 


2847 


67 


650 


81- 2913 


2980 


3047 


3'i4 


3181 i 3247 


3314 


3381 


3448 


3514 


67 


651 


8x- 3581 


3648 


3714 


3781 


3848 ! 3914 


3981 


4048 


4114 


4181 


67 


652 


8x- 4248 


43 >4 


4381 


4447 


45H 4581 


4647 


4714 


4780 


4847 


67 


J53 


81^4913 


4980 


5046 


5"3 


5179 5246 


5312 


5378 


5445 


55" 


66 


654 
655 


81- 5578 


5644 


57" 


5777 


5843 , 5910 


5976 


6042 


6109 


6175 


66 


81- 6241 


6308 


6374 


6440 


6506 


6573 


6639 


6705 


6771 


6838 


66 


656 


81- 6904 


6970 


7036 


7102 


7169 


7235 


7301 


7367 


7433 


7499 


66 


^l 


81- 7565 


7631 


7698 


7764 


7S30 


7896 


7962 


8028 


8094 


8160 


66 


658 


81- 8226 


8292 


8358 


8424 


8490 


8556 


8622 


8688 


8754 


8820 


66 


659 81- 8885 


8951 


9017 


9083 


9149 


9215 


9281 


9346 


9412 


9478 


66 


660 


81- 9544 


9610 


9676 


9741 


9807 


9873 


9939 


 






66 


660 


82- 














0004 


0070 


0136 


66 


661 


82- 0201 


0267 


0333 
09S9 


0399 


0464 


0530 


0595 


0661 


0727 


0792 


66 


662 


82- 0S58 


0924 


1055 


1 120 


1 186 


1251 


1317 


1382 


1448 


66 


5^3 


8a- 15 14 


1579 


1645 


1710 


1775 


184E 


1906 


1972 


2037 


2103 


65 


664 


82- 2168 


2233 


2299 


2364 


2430 


2495 


2560 


2626 


2691 


2756 


65 


6^5 


82- 2822 


2887 


2952 


3018 


3083 


3148 


3213 


3279 


3344 


3409 


65 


666 


82- 3474 


3539 


3605 


3670 


3735 


3800 


3865 


3930 


3996 


4061 


55 


/ 667 ' 83- 4126 


4191 


4256 


4321 


4386 


4451 


45 '6 


4581 


4646 


47" 


^J 


668 


82- 4776 


4841 


4906 


4971 


5036 


5101 


5166 


5231 

• 


5296 


5361 


65 
D 


N : 





I 


2 


3 


4 


5 


6 


7 


8 


9 






52 



MATHEMATICAL TABLES. 



N 
669 





I 


2 


3 


4 


S 


6 


7 


8 


9 


D 

65 


82- 5426 


5491 


5556 


5621 


5686 


5751 


5815 


5880 


5945 


6010 


670 


82- 6075 


6140 


6204 


6269 


6334 


6399 


6464 


6528 


6593 


6658 


65 


671 


82- 6723 


6787 


6852 


6917 


6981 


7046 


7111 


717s 


7240 


7305 


65 


672 


82- 7369 


7434 


7499 


7563 


7628 


7692 


7757 


7821 


7886 
8531 


7951 


65 


673 


82- 8015 


8080 


8144 


8209 


8273 
8918 


8338 


8402 


8467 


8595 


64 


674 


82- 8660 


8724 


8789 


8853 


8982 


9046 


9111 


9175 


9239 


64 


676 


82- 9304 


9368 


9432 


9497 


9561 


9625 


9690 


9754 


9818 


9882 


64 


676 


82- 9947 




















64 


676 


83- 


001 1 


0075 


0139 


0204 


0268 


0332 


0396 


0460 


0525 


64 


677 


83- 0589 


obSS 


0717 


0781 


084s 


0909 


0973 


1037 


X102 


1166 


64 


678 


83- 1230 


1294 


1358 


1422 


X486 


15 JO 


1614 


1678 


1742 


1806 


64 


679 


83- 1870 


1934 


1998 


2062 


2126 


2189 


2253 


2317 


2381 


2445 


64 


680 


83- 2509 


2573 


2637 


2700 


2764 


2828 


2892 


2956 


3020 


308^ 


64 


681 


83- 3147 


3211 


3275 


3338 


3402 


3466 


3530 - 


3593 


3657 


3721 


64 


682 


83- 3784 


3848 


3912 


3975 


4039 


4103 


4166 


4230 


4294 


4357 


64 


683 


83- 4421 


4484 


4548 


461 1 


4675 


4739 


4802 


4866 


4929 


4993 


64 


684 


83- 5056 


5120 


5183 


5247 


5310 


5373 


5437 


5500 


5564 


5627 


63 


685 


83- 5691 


5754 


5817 


5881 


5944 


6007 


6071 


6134 


6197 


6261 


63 


686 


83- 6324 


6387 


6451 
7083 


6514 


6577 


6641 


6704 


6767 


6830 


6894 


63 


687 


83- 6957 


7020 


7146 


7210 


7273 


7336 


7399 


7462 


7525 


63 


688 


83- 75»8 


7652 


7715 


7778 


7841 


7904 


7967 


8030 


8093 


8156 


63 


689 


83- 8219 


8282 


8345 


8408 


8471 


8534 


8597 


8660 


8723 


8786 


63 


690 


83- 8849 


8912 


8975 


9038 


9101 


9164 


9227 


9289 


9352 


9415 


63 


691 


83- 9478 


9541 


9604 


9667 


9729 


9792 


9855 


9918 


9981 




63 


691 


84- 








• •  • • 










0043 


63 


692 


84- 0106 


0169 


0232 


0294 


0357 


0420 


0482 


0545 


0608 


0671 


63 


693 


84- 0733 


0796 


0859 


0921 


0984 


1046 


1 109 


1172 


1234 


1297 


63 


694 


84- 1359 


1422 


1485 


1547 


1610 


1672 


1735 


1797 


i860 


1922 


63 


695 


84- 1985 


2047 


21 10 


2172 


2235 


2297 


2360 


2422 


2484 


2547 


62 


696 


84- 2609 


2672 


2734 


2796 


28^9 


2921 


2983 


3046 


3108 


3170 


62 


697 


84- 3233 


3295 


3357 
3p8o 


3420 


3482 


3544 


3606 


3669 


3731 


3793 


62 


698 


84- 3855 


3918 


4042 


4104 


4166 


4229 


4291 


4353 


4415 


62 


699 


84- 4477 


4539 


4601 


4664 


4726 


4788 


4850 


4912 
5532 


4974 
5594 


5036 


62 


700 


84- 5098 


5160 


5222 


5284 


5346 


5408 


5470 


5656 


62 


701 


84- 5718 


S780 


5842 


5904 


5966 


6028 


6090 


6151 


6213 


6275 


62 


702 


84- 6337 


6399 


6461 


6523 


6585 


6646 


6708 


6770 


6832 


6894 


62 


703 


84- 6955 


7017 


7079 


7141 


7202 


7264 


7326 


7388 


7449 


7511 


62 


704 


84- 7573 


7634 


7696 


7758 


7819 


7881 


7943 


8004 


8066 


8128 


62 


705 


84- 8i<89 


8251 


8312 


8374 


8435 


8497 


8559 


8620 


8682 


8743 
9358 


62 


706 


84- 8805 


8866 


8928 


8989 


9051 


9112 


9174 


9235 


9297 


61 


707 


84- 9419 


9481 


9542 


9604 


9665 


9726 


9788 


9849 


9911 


9972 


61 


708 


85- 0033 


0095 


0156 


0217 


0279 


0340 


0401 


0462 


0524 


0585 


61 


709 


85- 0646 


0707 


0769 


0830 


0891 


0952 
1564 


1014 


1075 


1 136 


1 197 


61 


710 


85- 1258 


1320 


1381 


1442 


1503 


1625 


1686 


1747 


1809 


61 


711 


85- 1870 


1931 


1992 


2053 


21 14 


2175 


2236 


2297 


2358 


2419 


61 


712 


85- 2480 


2541 


2602 


2663 


2724 


2785 


2846 


2907 


2968 


3029 


61 


713 


85- 3090 


3150 


3211 


^iP 


3333 


3394 


3455 


3516 


3577 


3637 


61 


714 

N 


85- 3698 


3759 


3820 


3881 


3941 


4002 


4063 


4124 


4185 

• 


4245 


61 
D 





I 


2 


3 


' 4 


s 


6 


7 


8 


9 



I' 



/ 



I.OGARITHMS OF NUMBERS. 




8 



43^7 44^S 4488 4549 

4974 5054 509s 5156 

5580 5640 5701 5761 

6185 624s ^306 6366 

678g 68so 6910 6970 



4610 4670 4731 

5216 5277 5337 

5822 5882 594J 

6427 6487 6548 

7031 7091 7152 



72' siS^^ 7J93 7453 
^^3 It !p7 8597 86S7 

7M I d. ^^^^ 9799 9S59 



75^3 
Si 16 

871S 

93^1 

99^^ 



7574 
8176 

8778 

9379 
9978 



86^ 
86^ 




7634 
8236 

8838 
9439 



7694 7755 

8297 8357 

8898 8958 

9499 9559 



7815 7875 

8417 8477 

9018 9078 

9619 9679 



0038 0098 0158 0218 0278 



86. i?^ 0996 lOSS 

^^^^ '594 les^ 



EE mO 

X7'4. 
^3x0 



0578 
1 176 

1773 
2370 

2966 



0637 0697 0757 

1236 1295 1355 

1833 1893 1952 

2430 2489 2549 

3025 3085 3144 



0817 0877 

H15 1475 

2012 2072 

2608 2668 

3204 3263 



7J4/S^'^^ 



sS^c^ A^3<^ 




9525 9584 964^ 9701 9760 



01 I I 0170 0228 0287 0345 

0696 0755 0813 0872 0930 

1281 1339 1398 1456 I5I5 

1865 1923 I98I 2040 2098 



S3 



D 



4792 4852 61 

5398 5459 61 

6003 6064 61 

6608 6668 60 

7212 7272 60 



60 
60 
60 
60 
60 
60 

60 
60 
60 
60 
60 



3620 3680 3739 3799 3858 59 

4214 4274 4333 4392 4452 59 

4808 4867 4926 4985 5045 59 

5400 5459 5519 5578 5637 59 

5992 6051 61x0 6169 6228 59 



6583 6642 6701 6760 6819 59 

7173 7232 7291 7350 7409 59 

7762 7821 7880 7939 7998 59 

8350 8409 8468 8527 8586 59 

8938 8997 9056 9114 9173 59 



59 
59 
59 

58 
58 



2448 2506 2564 2622 2681 58 

3030 3088 3146 3204 3262 58 

361 1 3669 3727 3785 3844 58 

4192 4250 4308 4366 4424 58 

4772 4830 4888 4945 5003 58 

5351 5409 5466 5524 5582 58 

5929 5987 6045 6102 6160 58 

6507 6564 6622 6680 6737 58 

7083 7141 7199 7256 7314 58 

7659 7717 7774 7832 7889 58 

8234 8292 8349 8407 8464 57 

8809 8866 8924 8981 9039 57 

9383 9440 9497 9555 9612 57 

9956 ••• 57 

0013 0070 0127 0185 57 

0528 0585 0642 0699 0756 57 

«; 6 7 8 Q D 



54 



MATHEMATICAL TABLES. 



N 



8 



D 



760 

761 
762 

763 
764 



765 

766 
767 
768 
769 



770 

771 
772 

773 
774 



88- 0S14 
88- 1385 
88- 1955 
88- 2525 
88- 3093 



0871 
1442 
2012 

2581 

3150 



0928 
1499 
2069 
2638 
3207 



0985 
1556 
2126 
2695 
3264 



1042 
1613 
2183 

2752 
3321 



1099 
1670 
2240 
2809 

3377 



1156 
1727 
2297 
2866 

3434 



1213 

1784 
2354 
2923 
3491 



1271 
1 841 
241 1 
2980 
3548 



1328 
1898 
2468 

3037 
3605 



88- 3661 
88- 4229 
88- 4795 
88- 5361 
88- 5926 



3718 
4285 

4852 

5418 

5983 



3775 
4342 
4909 

5474 
6039 



3832 

4399 

4965 

5531 
6096 



3888 

4455 
5022 

5587 
6152 



3945 
4512 

5078 

5644 
6209 



4002 
4569 

5135 
5700 

6265 



4059 
4625 

5192 

5757 
6321 



4115 
4682 

5248 

581 

637 



4172 

4739 

5305 
5870 

6434 



88- 6491 
88- 7054 
88- 7617 
88- 8179 
88- 8741 



6547 
7111 

7674 

8236 

8797 



6604 
7167 

7730 
8292 

8853 



6660 
7223 
7786 

8348 
8909 



6716 
7280 

7842 
8404 
8965 



6773 
7336 
7898 
8460 
9021 



6829 
7392 

8516 
9077 



6885 

7449 
801 1 

8573 
9134 



6942 

7505 
8067 

8629 

9190 



6998 
7561 
8123 
'8685 
9246 



57 
57 
57 
57 
57 



57 
57 
57 

57 
56 

56 
56 
56 
56 
56 



775 

776 
776 

777 
778 

779 



780 

781 
782 

783 
784 



785 

786 

787 
788 

789 

790 

791 
792 

793 
794 
794 



805 

N 



88- 9302- 

88- 9862 

89- 

89- 0421 

89- 0980 
89- 1537 



9358 
9918 

0477 
1035 
1593 



9414 
9974 

0533 
1091 

1649 



9470 9526 



OOJO 

0589 
1 147 
1705 



0086 
0645 

1203 
1760 



9582 9638 9694 9750 9806 



0I4I 
0700 

1259 
I8I6 



0197 
0756 

1314 

1872 



0253 
0812 

1370 
1928 



0309 
0868 

1426 

1983 



0365 

0924 
1482 

2039 



89- 209s 
89- 2651 

89- 3207 
89- 3762 
89- 4316 



2150 
2707 

3262 

3817 

4371 



2206 
2762 

3318 

3873 

4427 



2262 

2818 

3373 
3928 

4482 



2317 

2873 
3429 
3984 
4538 



2373 
2929 

3484 
4039 
4593 



2429 
2985 

3540 
4094 
4648 



2484 
3040 

3595 
4150 

4704 



2540 
3096 

3651 
4205 

4759 



2595 

3>5i 
3706 
4261 

4814 



89- 4870 
89- 5423 

89- 5975 
89- 6526 

89- 7077 



4925 

6030 
6581 

7132 



4980 

5533 
6085 

6636 

7187 



5036 
5588 
6140 
6692 
7242 



5091 

5644 
6195 
6747 
7297 



5146 
5699 
6251 
6802 
7352 



5201 

5754 
6306 

6857 
7407 



5257 
5809 
6361 

6912 
7462 



5312 
5864 
6416 
6967 

7517 



5367 
5920 

6471 

7022 

7572 



8g- 7627 
89- 8176 
89- 8725 
89- 9273 

89- 9821 

90- 



7682 
8231 
8780 
9328 

9875 



7737 
82S6 

8835 
9383 
9930 



7792 

8341 
8890 

9437 
9985 



7847 
8396 

8944 
9492 

0039 



795 90- 0367 0422 0476 0531 0586 

796 90- 0913 0968 1022 1077 1 131 

797 90- 1458 1513 1567 1622 1676 

798 90- 2003 2057 21 12 2166 2221 

799 90- 2547 2601 2655 2710 2764 



7902 
8451 
8999 
9547 



7957 
8506 

9054 
9602 



8012 
8561 
9109 
9656 



8067 
8615 
9164 
9711 



8122 
8670 
9218 
9766 



0094 0149 0203 0258 0312 



0640 
1186 

1731 

2275 
2818 



0695 
1240 

1785 
2329 

2873 



0749 

1295 
1840 

2384 
2927 



0804 

1349 
1894 

2438 

2981 



0859 
1404 
1948 
2492 
3036 



800 90- 3090 3144 3199 3253 3307 

801 90- 3633 3687 3741 3795 3849 

802 90- 4174 4229 4283 4337 4391 

803 90- 4716 4770 4824 4878 4932 

804 90- 5256 5310 5364 5418 5472 



90- 5796 5850 5904 5958 6012 



3361 

39<H 

4445 
4986 

5526 



3416 

3958 

4499 
5040 

5580 



3470 
4012 

4553 
5094 

5634 



3524 
4066 
4607 
5148 
5688 



3578 
4120 
4661 
5202 
5742 



56 
56 

56 
56 



56 
56 
56 
55 
55 



55 

55 
55 
55 

ii 

55 
55 
55 
55 
55 
55 



55 
55 
54 
54 
54 



54 
54 
54 
54 
54 



6066 6119 6173 6227 6281 54 
^ 6 7 S I) 



LOGARITHMS OF NUMBERS. 



55 



N 



8 



D 



'So6 

■807 

808 

;809 



90- 6335 6389 

90- 6874 6927 

90- 7411 7465 

90- 7949 8002 



6443 6497 6551 

6981 7035 7089 

75*9 7573 7626 

S056 81 10 8163 



6604 

7143 
7680 

8217 



6658 
7196 

7734 
8270 



6712 
7250 

7787 
8324 



6766 

7304 

7841 

8378 



6820 
7358 
7«95 
8431 



810 

811 
812 

8f2 

814 



90- 8485 8539 

9&- 9021 9074. 

90- 9556 9610 

y*^ «>•••• m m 9 m m m 

91- 0091 0144. 

91- 0624 0678 



8592 8646 
9128 9181 
9663 9716 



0197 0251 
0731 0784 



8699 

9235 
9770 

0304 
0838 



8753 
9289 
9823 

0358 
0891 



8807 
9342 
9877 

041 1 
0944 



8860 
9396 
9930 

0464 
0998 



8914 

9449 
9984 

0518 
105 1 



8967 
9503 

0037 
0571 
1104 



54 
54 
54 

54 



54 
54 
54 
53 
53 
53 



815 

816 

817 

:8ig 

819 



91- 1158 121 1 1264. 

91-1690 1743 '797 

91- 2222 2275 ^3^^ 

91- 2753 2806 2»59 

9»- 3284 3337 3390 



1317 
1850 

2381 

29<3 
3443 



1371 

1903 

2435 
2966 

3496 



1424 
1956 
2488 
3019 

3549 



1477 
2009 

2541 

3072 

3602 



1530 
2063 

2594 
3125 
3655 



1584 
2116 
2647 

3178 
3708 



1637 
2169 

2700 

3231 
3761 



820 

821 

822 
823 
824 



91- 
91- 
91- 
91- 
91- 



3814 

4343 
4872 

S40O 
5927 



3867 

439^ 

4925 

5453 
5980 



3920 

4449 
4977 

6033 



3973 
4502 

5030 

5558 
6085 



4026 

4555 
5083 
561 1 

6138 



4079 
4608 

5136 
5664 

6191 



4132 
4660 

5189 
5716 

6243 



4184 

4713 
5241 

5769 
6296 



4237 
4766 

5294 

5822 

6349 



4290 
4819 

5347 

5875 
6401 



825 

826 

827 i 

828 

829 



91- 
91- 

9X- 
91- 
91- 



6454 
6980 
7506 
8030 

8555 



6507 
7033 

7558 
8083 

8607 



6559 
7085 

761 1 

8x35 
8659 



6612 

7138 
7663 
8188 
8712 



6664 
7190 
7716 
8240 
8764 



6717 

7243 
7768 

8293 
8816 



6770 

7295 
7820 

8345 
8869 



6822 
7348 
7873 

8397 
8921 



6S75 
7400 
7925 
8450 

8973 



6927 

7453 
7978 

8502 

9026 



830 ! 9X-9078 9130 
831 91- 9601 9653 

831 92- 

832 92- 0123 0176 

833 92- 0645 0697 

834 9a- 1166 1218 



9183 
9706 



0228 

0749 
1270 



9235 
9758 

0280 
0801 
1322 



9287 
9810 

0332 

0853 
1374 



9340 
9862 

0384 
0906 
1426 



9392 
9914 

0436 
0958 
1478 



9444 
9967 

0489 

lOIO 

1530 



9496 9549 



0019 

0541 
1062 

1582 



0071 

0593 
1 1 14 

1634 



835 

836 

838 
839 



92- 1686 
92- 2206 
9a- 2725 
92- 3244. 
92- 3762 



1738 
2258 

^777 
3296 

3814 



1790 
2^10 
2829 

m 



1842 
2362 
2881 

3399 
3917 



1894 
2414 

2933 
3451 
3969 



1946 
2466 

2985 

3503 
4021 



1998 
2518 

3037 

3555 
4072 



2050 
2570 
3089 
3607 
4124 



2102 
2622 
3140 

3658 
4176 



2154 
2674 
3192 

3710 
4228 



840 

841 
842 

,843 
' 844 



9»- 4279 
92- 4796 

92- 5312 

92- 5828 

92- 6342 



4848 
53^4 
5879 
6394 



4383 
4899 

5415 
593 « 
6445 



4434 
4951 
5467 
5982 
6497 



4486 

5003 
5518 

6034 
6548 



4538 
5054 
5570 
6p85 
6600 



4589 
5106 

5621 

6137 
6651 



4641 

5157 

5673 
6188 

6702 



4693 
5209 

5725 
6240 

6754 



4744 
5261 

5776 
6291 
6805 



845 

1846 

I 847 
'848 

I 849 



92- 6857 
92- 7370 
92- 78S3 
92- 8396 
92- 8908 



6908 
7422 

7935 
8447 
8959 



6959 

7473 
7986 

8498 

9010 



701 1 

7524 

8037 

8549 
9061 



7062 

7576 
8088 
8601 
9112 



7114 
7627 
8140 
8652 
9163 



7165 
7678 
8191 

8703 
9215 



7216 

7730 
8242 

8754 
9266 



7268 

7781 
8293 
8805 

9317 



73^9 

7832 

8345 
8857 

9368 




93- 9419 
92- 9930 



9470 
9981 



9521 9572 9623 



9674 9725 9776 9827 9879 



7 



8 



53 
53 
53 

53 
53 



53 
53 
53 
53 
53 



53 
53 

52 
52 
52 



52 
52 
52 
52 
52 
52 



52 
52 
52 
52 

52 



52 
52 
52 
51 
51 



51 
51 
51 
51 
51 



51 
51 

D 



S6 



MATHEMATICAL TABLES. 



N 





I 


2 


3 


4 


5 


6 


7 


8 9 


D 


851 


93- 




0032 


0083 


0134 


0185 


0236 


0287 


0338 0389 
0S47 0898 


51 


852 


93- 0440 


0491 


0542 


0592 


0643 


0694 


0745 


0796 


51 


853 


93- 0949 


IOCX> 


105 1 


1 102 


1153 


1203 


1254 


1305 


1356 1407 
1865 1915 


SI 


854 


93- 1458 


1509 


1560 


1610 


1661 


1712 


1763 


1814 


51 
51 


856 


93- 1966 


2017 


2068 


2118 


2169 


2220 


2271 


2322 


2372 2423 


856 


93- 2474 


2524 


2575 


2626 


2677 


2727 


2778 


2829 


2879 2930 


SI 


857 


93- 2981 


3031 


3082 


3133 


3183 


3234 


3285 


3335 


3386 3437 
3892 3943 
4397 4448 


SI 


858 


93- 3487 


3538 


3589 


3639 


3690 


3740 


3791 


3841 


51 


859 


93- 3993 


4044 


4094 


4145 


4195 


4246 


4296 


4347 


SI 


860 


93- 4498 


4549 


4599 


4650 


4700 


4751 


4801 


4852 


4902 4953 


50 


861 


93- 5003 


5054 


5104 


5154 


5205 


5255 


5306 
5809 


5356 


5406 5457 


50 


862 


93- 5507 


5558 


5608 


5658 


5709 


5759 


5860 


5910 5960 


50 


863 


93- 601 I 


6061 


6111 


6162 


6212 


6262 


6313 
6815 


6363 
6865 


6413 6463 


50 


864 


93- 6514 


6564 


6614 


6665 


6715 


6765 


6916 6966 


50 


865 


93- 7016 


7066 


7117 
7618 


7167 


7217 


7267 


7317 


7367 


7418 7468 


50 


866 


93- 75>8 


7568 


7668 


7718 


7769 
8269 


7819 


7869 


7919 7969 


50 


867 


93- 8019 


8069 


8119 


8169 


8219 


8319 


8370 


8420 8470 


50 


868 


93- 8520 


8570 


8620 


8670 


8720 


8770 


8820 


8870 


8920 8970 


so 


869 


93- 9020 


9070 


9120 


9170 


9220 


9270 


9320 


9369 


9419 9469 


50 


870 


93- 9519 


9569 


9619 


9669 


9719 


9769 


9819 


9869 


9918 9968 


50 


871 


94- 0018 


0068 


0118 


0168 


0218 


0267 


0317 
0815 


0367 
0865 


0417 0467 


50 


872 


94- 0516 


0566 


0616 


0666 


0716 


0765 


0915 0964 


50 


873 


94- 1014 


1064 


1 1 14 


1 163 


1213 


1263 


1809 


1362 


1412 1462 


50 


874 


94- 1511 


1561 


1611 


1660 


1710 


1760 


1859 


1909 1958 


50 


875 


94- 2008 


2058 


2107 


2157 


2207 


2256 


2306 
2801 


2355 


2405 2455 


50 


876 


94- 2504 


2554 


2603 


2653 


2702 


2752 


2851 


2901 2950 


50 


^77 
878 


94- 3000 


3049 


3099 


3148 


3198 


3247 


3297 


3346 


3396 3445 


49 


94- 3495 


3544 


408!; 


3^43 


3692 


3742 


3791 


3841 


3890 3939 


49 


879 


94- 3989 


4038 


4137 


4186 


4236 


4285 


4335 


4384 4433 


49 


880 


94- 4483 


4532 


4581 


4631 


4680 


4729 


4779 


4828 


4877 4927 


49 


881 


94- 4976 


5025 


5074 


5124 


5>73 


5222 


5272 


5321 


5370 5419 
5862 59 r2 


49 


882 


94- 5469 


5518 


5567 


5616 


5665 


5715 


5764 


5813 


49 


883 


94- 5961 


6010 


6059 


6108 


6157 


6207 


6256 


6305 


6354 6403 
6845 6894 


49 


884 


94- 6452 


6501 


6551 


6600 


6649 


6698 


6747 


6796 


49 
49 


885 


94- 6943 


6992 


7041 


7090 


7140 


7189 


7238 


7287 


7336 7385 


886 


94- 7434 


7483 


7532 


7581 


7630 


7679 


7728 


7777 


7826 7875 


49 


887 


94- 7924 


7973 


8022 


8070 


8119 


8168 


8217 


8266 


8315 8364 
8804 8853 


49 


888 


94- 8413 


8462 


8511 


8560 


8609 


8657 


8706 


8755 


49 


889 


94- 8902 


8951 


8999 


9048 


9097 


9146 


9195 


9244 


9292 9341 


49 


890 


94- 9390 


9439 


9488 


9536 


9585 


9634 


9683 


9731 


9780 9829 


49 


891 


94- 9878 


9926 


9975 














49 


891 


95- 






0024 


0073 


OI2X 


0170 


0219 


0267 0316 
0754 0803 


49 


892 


95- 0365 
95- 0851 


0414 


0462 


0511 


0560 


0608 


0657 


0706 


49 


893 


0900 


0949 


0997 


1046 


1095 


"43 


1 192 


1240 1289 


49 


894 


95- 1338 


1386 


1435 


1483 


1532 


1580 


1629 


1677 


1726 1775 


49 


896 


95- 182J 
^ 95- 2308 


1872 


1920 


1969 


2017 


2066 


21 14 


2163 


22 I I 2260 


48 


896 


2356 


^?5 


2453 


2502 


2550 


2599 


2647 


2696 2744 


48 


897 

N 


95- 2792 


2841 


2889 


2938 


2986 


3034 


3083 


3131 


3180 3228 


48 





I 


2 


3 


4 


5 


6 


7 


8 9 


D 



LOGARITHMS OF NUMBERS. 



57 



i N 



8 



D 

48 
48 

48 
48 
48 
48 
48 



898 95- 3276 3325 3373 3421 3470 

899 ! 95- 3760 3808 3856 3905 3953 



3518 
4001 



3566 
4049 



361. 
409^ 



3663 
4146 



37" 
4194 



900 ' 95- 4243 4291 4339 

901 I 95- 4725 4773 4821 

95- 5207 5255 5303 

95- 5688 5736 5784 

95- 6168 6216 6265 



902 

903 
904 



905 

906 

,907 

I 908 

909 



4387 
4869 

5351 
32 

313 



4435 
4918 

5399 
5880 

6361 



95- 6649 
95- 7128 

95- 7607 
95- 8086 

95- 8564 



6697 

7176 

7655 
8134 
8612 



6745 
7224 

7703 
8181 

8659 



6793 
7272 

7751 
8229 

8707 



6840 
7320 

7799 
8277 
8755 



4484 
4966 

5447 
5928 
6409 



4532 
5014 

5495 
5976 

6457 



4580 
5062 

5543 
6024 

6505 



4628 
5110 

5592 
6072 

6553 



4677 

5158 
5640 

6120 

6601 



6888 

7368 

7847 
8325 
8803 



6936 
7416 
7894 

8373 
8850 



6984 
7464 
7942 
8421 
88q8 



7032 
7512 
7990 
8468 
8946 



7080 

7559 
8038 

8516 

8994 



48 
48 

48 
48 
48 

48 
48 
48 
48 
48 
47 



910 

911 
! 912 
912 

913 
914 



95- 9041 

95- 9518 

95-9995 

96- 

96- 0471 

96- 0946 



9089 
9566 

0042 
0518 
0994 



9137 
9614 

0090 
0566 
1041 



9185 
9661 

0138 
0613 
1089 



9232 
9709 

0185 
066 X 
1136 



9280 
9757 

0233 
0709 

1184 



9328 
9804 

•  • • «  

0280 
0756 
1231 



9375 
9852 

0328 
0804 
1279 



9423 
9900 

0376 
0851 
1326 



9471 
9947 

Q423 
0899 

1374 



916 I 96- 1421 



916 
917 
918 
919 



96- 1895 
96- 2369 
96- 2843 
9&- 33^6 



1469 

1943 
2417 

2890 
3363 



1516 
1990 
2464 

2937 
3410 



1563 
2038 

251 1 

2985 

3457 



1611 
2085 

2559 
3032 

3504 



1658 
2132 
2606 

3079 
3552 



1706 
2180 

2653 
3126 

3599 



1753 
2227 

2701 

3174 
3646 



1801 
2275 
2748 
3221 

3693 



1848 
2322 

2795 
3268 

3741 



47 
47 
47 
47 
47 



920 

921 
922 

923 
924 



96-3788 
96- 4260 

96-4731 
96- 5202 

96- 5672 



3835 
4307 
4778 
5249 
5719 



3882 

4354 
4825 
5296 
5766 



3929 
4401 

4872 

5343 
5813 



3977 
4448 

4919 
5390 
5^ 



4024 

4495 
4966 

5437 
5907 



4071 
4542 
5013 
5484 
5954 



4118 
4590 
5061 

5531 
6001 



4165 

4637 
5108 

6048 



4212 i 47 



4684 

5155 
5625 
6095 



47 
47 
47 
47 



925 

926 

927 
928 

929 



930 

1931 

' 932 

933 

933 

.934 



i" 



985 

936 
937 
938 
939 



940 

941 
942 
943 

IT 



96- 6142 
96- 661 I 
96- 7080 
96- 754S 
96- 8016 



6189 
6658 
7127 

7595 
8062 



6236 
6705 

7173 
7642 

8109 



6283 
6752 
7220 
7688 
8156 



6329 
6799 
7267 

7735 
8203 



6376 
6845 

7314 
7782 

8249 



6423 
6892 

7361 
7829 
8296 



6470 

6939 
7408 

7875 
8343 



6517 
6986 

7454 
7922 

8390 



6564 

7033 
7501 

7969 

8436 



96-8483 
96- 8950 
96- 9416 

96- 9882 

97- 

97- 0347 



8530 
8996 

9463 
9928 



8576 

9043 
9509 

9975 



0393 0440 



8623 
9090 

9556 

0021 

0486 



8670 

9136 
9602 

0068 
0533 



8716 

9183 
9649 

01 14 
0579 



8763 
9229 

9695 

0161 
0626 



8810 
9276 
9742 

0207 
0672 



8856 

9323 
9789 

0254 
0719 



8903 

9369 
9835 

0300 
0765 



97- 0812 
97- 1276 
97- 1740 
97- 2203 
97- 2666 



0858 
1322 
1786 

2249 
2712 



0904 

1369 
1832 

2295 
2758 



0951 
1415 

1879 
2342 
2804 



0997 
1461 
1925 
2388 
2851 



1044 
1508 
1971 

2434 
2897 



1090 

>554 
2018 

2481 
2943 



"37 
1601 

2064 

2527 
2989 



1x83 
1647 
21 10 

2573 
3035 



97- 3128 
97- 3590 
97- 4051 
97- 4512 



3174 
3636 

4097 
4558 



3220 
3682 

4143 
4604 



3266 
3728 
4189 
4650 



3313 

3774 

4235 
4696 



3359 
3820 

4281 
4742 



3405 
3866 

4327 
4788 



3451 
3913 
4374 
4834 



3497 

3959 
4420 
4880 

8 



1229 

1693 
2157 

2619 

3082 

3543 
4005 

4466 

4926 



47 
47 
47 
47 
47 



47 

47 

47 

47 

47 
46 



46 

46 
46 
46 
46 

46 
46 
46 

ii 

D 



- "-It. '-/*f /■ 



•  58 



MATHEMATICAL TABLES. 



N 

944 





I 


2 


3 


4 


5 


6 


7 


8 


9 


D 

46 


97- 4972 


5018 


5064 


5110 


5156 


5202 


5248 


5294 


5340 


5386 


945 


97- 5432 


5478 


5524 


5570 


5616 


5662 


5707 


5753 


• 

5799 


5845 


46 


946 


97- 5891 


5937 


5983 


6029 


6075 


6121 


6167 


6212 


6258 


6304 


46 


947 


97- 6350 
97- 6808 


6396 


6442 


64S8 


6533 


6579 


6625 


6671 


6717 


6763 


46 


948 


6854 


6900 


6946 


6992 


7037 


7083 


7129 


7175 


7220 


46 


949 


97- 7266 


7312 


7358 


7403 


7449 


7495 


7541 


7586 


7632 


7678 


46 


960 


97- 7724 


7769 


7815 


7861 


7906 


7952 
8409 


7998 
8454 


8043 


8089 


8135 


46 


951 


97- 8i8i 


8226 


8272 


8317 


8363 


8500 


8546 


8591 


46 


952 


97- 8637 


8683 


8728 


8774 


8819 


8865 


891 1 


8956 


9002 


9047 


46 


953 


97-9093 


9138 


9184 


9230 


9275 


9321 


9366 


9412 


9457 


9503 


46 


954 


97- 9548 


9594 


9639 


9685 


9730 


9776 


982 X 


9867 


9912 


9958 


46 


966 


98- 0003 


0049 


0094 


0140 


D185 


0231 


0276 


0322 


0367 
082 X 


^l^ 


45 


956 


98- 0458 


0503 


0549 


0594 


0640 


0685 


0730 


0776 


0867 


45 


957 


98- 0912 


0957 


1003 


1048 


1093 


"39 


1x84 


1229 


1275 


1320 


45 


958 


98- 1366 


X411 


1456 


1 501 


1547 


1592 


1637 


1683 


1728 


1773 


45 


959 


98- 1819 


1864 


1909 


1954 


2000 


2Q45 


2090 


2135 


2181 


2226 


45 


960 


98- 2271 


2316 


2362 


2407 


2452 


2497 


2543 


2588 


2653 
3085 


2678 


45 


961 


98- 2723 


2769 


2814 


2859 


2904 


2949 


2994 


3040 


3130 


45 


962 


98- 3175 


3220 


3265 


33 JO 


3356 
3807 


3401 


3446 


3491 


3536 


3581 


45 


963 


98- 3626 


3671 


3716 


3762 


3852 


3897 


3942 


3987 


40J2 


45 


964 


98- 4077 


4122 


4167 


4212 


4257 


4302 


4347 


4392 


4437 


4482 


45 


965 


98- 4527 


4572 


4617 


4662 


4707 


4752 


4797 


4842 


4887 


4932 


45 


966 


98- 4977 


5022 


5067 


5112 


5157 


5202 


5247 


5292 


5337 


5382 


45 


967 


98- 5426 


5471 


5516 


5561 


5606 


5651 


5696 


5741 


5786 


5830 


45 


968 


98- 5875 


5920 


5965 


6010 


6055 


6100 


6144 


6189 


6234 


6279 


45 


969 


98- 6324 


6369 


6413 


6458 


6503 


6548 


6593 


6637 


6682 


6727 


45 


970 


98- 6772 


6817 


6S61 


6906 


6951 


6996 


7040 


7085 


7130 


7175 


45 


971 


98- 7219 


7264 


7309 


7353 
7800 


7398 


7443 


7488 


7532 


7577 


7622 


45 


972 


98- 7666 


77" 


7756 


7845 


7890 


7934 


7979 


8024 


8068 


45 


973 


98- 81 13 


8157 


8202 


8247 


8291 


8336 


838X 


8425 


8470 


8514 


45 


974 


98- 8559 


8604 


8648 


8693 


8737 


8782 


8826 


8871 


8916 


8960 


45 


975 


98- 9005 


9049 


9094 


9138 


9185 


9227 


9272 


9316 


9361 


9405 


45 


976 


98- 9450 


9494 


9539 


9583 


9628 


9672 


9717 


9761 


9806 


9850 


44 


977 


98- 9895 


9939 


9983 


• •  •  














44 


977 


99- 






0028 


0072 


0117 


0161 


0206 


0250 


0294 


44 


978 


99- 0339 


0383 


0428 


0472 


0516 


0561 


0605 


0650 


0694 


0738 


44 


979 


99- 0783 


0827 


0871 


0916 


0960 


1004 


1049 


1093 


1137 


1 182 


44 


980 


99- 1226 


1270 


I3J5 


1359 


1403 


1448 


1492 


1536 


1580 


1625 


44 


981 


99- 1669 


1713 


1758 


1802 


1846 ! 1890 


1935 


1979 


2023 


2067 


44 


982 


99- 21 I I 


2156 


2200 


2244 


2288 


2333 


2377 


2421 


2465 


2509 


44 


983 


99- 2554 
99- 2^95 


2598 


2642 


2686 


2730 


2774 


2819 


2863 


2907 


2951 


44 


984 


3039 


3083 


3127 


3172 


3216 


3260 


3304 


3348 


3392 


44 


985 


99- 3436 


3480 


3524 


3568 


3613 


3657 


3701 


3745 


3789 


3833 


1 
44 


986 


99- 3877 


3921 


3965 


4009 


4053 


4097 


4141 


4185 


4229 


4273 


44 


987 


99- 4317 


4361 


4405 


4449 


4493 


4537 


4581 


4625 


4669 


4713 


44 


988 


99- 4757 


4801 


4845 


4889 


4933 


4977 


5021 


5065 


5108 


5152 


44 


989 

N 


99- 5196 


5240 


5284 


5328 


5372 


5416 


5460 


5504 


5547 


5591 


44 
D 





I 


2 


3 


4 


5 


6 


7 


8 


9 



LOGARITHMS OF NUMBERS. 



59 



N 





I 


2 


3 


4 


5 


6 


7 


8 


9 


D 


990 


9^- 5635 


5679 


5723 


5767 


5811 


5854 


5898 


5942 


5986 


6030 


44 


991 


9^- 6074. 


6117 


6161 


6205 


6249 


6293 


6337 


6380 


6424 


6468 


44 


992 


99- 6512 


6555 


6599 


6643 


6687 


6731 


6774 


6818 


6862 


6906 


44 


993 


9Q- 694.9 


6993 


7037 


7080 


7124 


7168 


7212 


7255 


7299 


7343 


44 


994 


99- 73S^ 


7430 
7867 


7474 
7910 


7517 

7954 


7561 
7998 


7605 
8041 


7648 


7692 


7736 


7779 


44 
44 


995 


99- 7823 


8085 


8129 


8172 


8216 


996 


99- 8259 


8303 


8347 


83QO 


8434 


8477 


8521 


8564 


8608 


8652 
9087 


44 


997 


99- 8695 


8739 


8782 


8826 


8869 


8913 


8956 


9000 


9043 


44 


99» 


99- 9131 


9174 


9218 


9261 


9305 


93*8 


9392 


9435 


9479 


9522 


44 


999 


99- 9565 


9609 


9652 


9696 


9739 


9783 


9826 


9870 


9913 


9957 


43 
D 


N 





I 


2 


3 


4 


5 


6 


7 


8 


9 



/ 



6o 



MATHEMATICAL TABLES. 



TABLE No. IL— HYPERBOLIC LOGARITHMS OF NUMBERS 

FROM 1. 01 TO 30. 



Number. 


Logarithm. 


Number. 


Logarithm. 


Number. 


Logarithm. 


Number. 


Logarithm. 


I.OI 
1.02 
1.03 
1.04 

1.05 


.0099 
.0198 
.0296 
.0392 
.0488 


1.36 

1-37 

1.38 

1-39 
1.40 


•3075 
.3148 
.3221 

.3293 
•3365 


I.7I 
1.72 

1-73 

1.74 

1 1.75 


•5365 

.5423 
.5481 

•5539 
•5596 


2.06 
2.07 
2.08 
2.09 
2.10 


.7227 

•7275 
•7324 

•7372 

.7419 


1.06 
1.07 
1.08 
1.09 
1. 10 


.0583 
.0677 

.0770 

.0862 

■0953 


1.41* 
1.42 

1-43 
1.44 

1.45 


.3436 
.3507 

.3577 
.3646 

•3716 


1.76 
1.77 
1.78 
1.79 
1.80 


.5653 

.5710 
.5766 

.5822 
.5878 


2. II 
2.12 
2.13 
2.14 

2^15 


.7467 
•7514 
.7561 
.7608 

.7655 


1. 11 

1. 12 

^•13 
1. 14 

^•15 


.1044 

.1133 
.1222 

.1310 

.1398 


1.46 

1.47 
1.48 

1.49 

1.50 


.3784 

.3853 
.3920 

.3988 
•4055 


1.81 
1.82 

1.83 
1.84 

1.85 


.5933 
.5988 
.6043 
.6098 
.6152 


2.16 
2.17 
2.18 
2.19 
2.20 


.7701 
.7747 

•7793 

.7839 
.7885 


1. 16 
1.17 

1. 18 

1. 19 
1.20 


.1484 

.1570 
.1655 
.1740 
.1823 


I-5I 
1-52 

1-53 
1^54 
1.55 


.4121 
.4187 

•4253 
.4318 

■4383 


1.86 
1.87 
1.88 
1.89 
1.90 


.6206 
.6259 

.6313 
.6366 

.6419 


2.21 
2.22 
2.23 
2.24 

2.25 


•7930 

.7975 
.8020 

.8065 

.8109 


1. 21 
1.22 
1.23 
1.24 

1.25 


.1906 
.1988 
.2070 
.2151 
.2231 


1.56 

1-57 

1.58 

1.59 
1.60 


.4447 
.4511 

•4574 

•4637 
.4700 


1.91 
1.92 

1.93 
1.94 

1.95 


.6471 

•6523 

•6575 
.6627 

.6678 


2.26 
2.27 
2.28 
2.29 
2.30 


.8154 
.8198 
.8242 
.8286 

•8329 


1.26 
1.27 
1.28 
1.29 
1.30 


.2311 
.2390 
.2469 
.2546 
.2624 


1.61 
1.62 
1.63 
1.64 

1.65 


.4762 
.4824 
.4886 

.4947 
.5008 


1.96 

1.97 
1.98 
1.99 

2.00 


.6729 
.6780 
.6831 
.6881 
.6931 


2.31 
2.32 

2.33 
2.34 
2.35 


.8372 
.8416 

.8458 

.8502 

.8544 


131 
1.32 

1.33 
1.34 
I.3S 


.2700 
.2776 
.2852 
.2927 
.3001 


1.66 
1.67 
1.68 
1.69 
1.70 


.5068 
.5128 
.5188 

.5247 
.5306 


2.01 
2.02 
2.03 
2.04 
2.05 


.6981 

•7031 
.7080 

.7129 
.7178 


2.36 

2.37 
2.38 

2.39 
2.40 


.8587 
.8629 

.8671 

•8713 

.8755 



j 



HYPERBOLIC LOGARITHMS OF NUMBERS 



6l 



1 

4 


!f umber. 


Logarithm. 


Number. 


Logarithm. 


Number. 


Logarithm. 


1 
Number. 


Logarithm. 




2.41 


.8796 


2.8l 


1.0332 


3.21 


I.1663 


3.61 


1.2837 


2.42 


.8838 


2.82 


1.0367 


3.22 


1. 1694 


3.62 


1.2865 


2.43 


.8879 


2.83 


1.0403 


3.23 


I.I725 


3.63 


1.2892 




2.44 .8920 


2.84 


1.0438 


3.24 


I.1756 


3-64 


1.2920 




2.45 .8961 


2.85 


1.0473 


3.25 


I.I787 


3.65 


1.2947 




2.46 .9002 


2.86 


1.0508 


3.26 


1.1817 


3.66 


1.2975 




2.47 .9042 


2.87 


1-0543 


3-27 


1. 1848 


3.67 


1.3002 


2.48 1 -9083 


2.88 


1.0573 


328 


1. 1878 


3.68 


1.3029 


2.49 1 .9123 


2.89 


I.0613 


3.29 


1. 1 909 


3.69 


1.3056 




2.50 -9163 


2.90 


1.0647 


3.30 


1-1939 


370 


1.3083 




2.51 


.9203 


2.91 


1.0682 


3.31 


1. 1969 


3.71 


1.3110 


2-5* 


-9243 


2.92 


1.0716 


3.32 


1. 1999 


372 


I.3137 


*-53 


.92^2 


2.93 


1.0750 


3-33 


1.2030 


3.73 


I.3164 




2.54 


.9322 


2.94 


1.0784 


3-34 


1.2060 


3.74 


I.319I 




2-55 


.9361 


2.95 


i.o8i8 


3.35 


1.2090 


3.75 


1.3218 




2.56 


.9400 


2.96 


1.0852 


3.36 


1.2119 


3.76 


1.3244 


2.57 


•9439 


2.97 


1.0886 


3-37 


1. 2149 


3.77 


I.3271 


2.58 


.9478 


2.98 


1.0919 


3.38 


I.2179 


3.78 


1.3297 


1 2.59 


.9517 


2.99 


1.0953 


3-39 


1.2208 


3.79 


1.3324 




2.60 


-9555 


3.00 


1.0986 


3.40 


1.2238 


3.80 


1-3350 




2.61 


-9594 


3.01 


1.1019 


3.41 


1.2267 


3.81 


1.3376 




2.62 


.9632 


3.02 


1. 1053 


3.42 


1.2296 


3.82 


1.3403 




2.63 


.9670 


303 


1. 1 086 


3-43 


1.2326 


383 


1.3429 




2.64 


.9708 


304 


1.1119 


3-44 


1.2355 


3.84 


1-3455 




2.65 


.9746 


3-05 


1.1151 


3.45 


1.2384 


3.85 


1.3481 




2.66 


.9783 


3.06 


1.1184 


3.46 


1.2413 


3.86 


1-3507 


2.67 


.9821 


3-07 


1.1217 


3.47 


1.2442 


3.87 


1-3533 


2.68 


.9858 


3.08 


1. 1249 


3.48 


1.2470 


3.88 


1.3558 


1 2-69 


.9895 


.309 


1.1282 


3-49 


1.2499 


3.89 


1-3584 




2.70 


-9933 


3.10 


1.1314 


350 


1.2528 


3.90 


1.3610 




2.71 


.9969 


3" 


1. 1346 


3.51 


1.2556 


3-91 


1.3635 




2.72 1.0006 


3.12 


1.1378 


352 


1.2585 


3.92 


1.3661 




2.73 1.0043 


313 


1.1410 


3.53 


1. 2613 


3.93 


1.3686 




2.74 


1.0080 


3.14 


1.1442 


3.54 


1. 2641 


3.94 


1.3712 




2.75 


1.0116 


315 


1.1474 


3.55 


1.2669 


3.95 . 


1.3737 




^ 2.76 


1-0152 


3.16 


1. 1506 


3.56 


1.2698 


3.96 


1.3762 




2.77 


1.0188 


3-17 


1.1537 


3.57 


1.2726 


3-97 


1.3788 




2.78 


1.0225 


3.18 


1.1569 


3.58 


1.2754 


3.98 


1.3813 




2.79 1.0260 


319 


1. 1600 


3-59 


1.2782 


3-99 


1.3838 




2.80 


1.0296 


3.20 


1.1632 


3.60 


1.2809 


4.00 


1.3863 



! 



MATHEMATICAL TABLES. 



Logiiithm. 


I Numbe. 


Logarithm. 


Numbn. 


L.«lfithlB. 


Numbn. 


l.:g..ilhn.- 


1.3888 


4.41 


1.4839 


4.8. 


1-5707 


S-2I 


.-6506 


'■39'3 


1 4-42 


..4861 


4.82 


1-5728 


S-22 


1-6525 


'■3938 


i 4-43 


r.4884 


4.83 


1-5748 


5-23 


.-65.4 


1.3962 


; 4-44 


1.4907 


4.84 


1.5769 


S-24 


.6563 


1-3987 


,4.45 


1.4929 


4.85 


1.5790 


5-25 


.-6582 


1. 401 2 


, 4.46 


1-495' 


4.86 


1.5S.0 


5-26 


.-6601 


1.4036 


! 4-47 


1.4974 


4.87 


..583. 


5-27 


1-6620 


..4o6r 


4.48 


1.4996 


4.88 


1-5851 


5.28 


1-6639 


1.408s 


4.49 


I.50I9 


4.89 


..5872 


5.29 


1-6658 


1.4110 


1 4.50 


1.5041 


4.90 


1.5892 


5.30 


1-6677 


1.4134 


! 4-51 


1.5063 


4.91 


1-5913 


5-31 


1-6696 


i-4'S9 


; 4.52 


1.5085 


4.92 


1-5933 


S-32 


I-67I5 


1.4183 


i 453 


1.5107 


4.93 


1-5953 


5-33 


1-6734 


1.4207 


 4-54 


1.5129 


4-94 


1-5974 


S-34 


1-6752 


1.4231 


4.55 


1.5151 


4.95 


1-5994 


5-35 


1-6771 


I-42SS 


i 4.56 


i.S"73 


4.96 


1-6014 


5-36 


1-6790 


1.4279 


, 4.57 


'■S'95 


4.97 


1-6034 


S-37 


1-6808 


1.4303 


,4.58 


1-5217 


4.98 


1-6054 


5-38 


1-6827 


1-4327 


1 4-59 


1.5239 


499 


1-6074 


5-39 


1-6845 


I-43S1 


I4.60 


1-526. 


5.00 


1-6094 


5-40 


I-6S64 


•■4375 


, 4.61 


1.5282 


! 5-01 


1-6.14 


S-41 


1-6882 


1.4398 


4.62 


1-5304 


5.02 


1-6134 


5-42 


1-690, 


1.4422 


4.63 


1.5326 


5-03 


1-6154 


5-43 


I-69I9 


1.4446 


 4-64 


1.5347 


5.04 


.-6,74 


5-44 


1-6938 


1.4469 


, 4-65 


1.5369 


S-OS 


.-6194 


S-45 


1-6956 


1-4493 


4.66 


■■5390 


5.06 


1-6214 


5-46 


1-6974 


1.4516 


4.67 


1.5412 


5.07 


.6233 


5-47 


1-6993 


1.4540 


4.68 


1.S433 


5.08 


.-6253 


S-48 


1-7011 


1.4563 


4.69 


1.5454 


5.09 


1-6273 


5-49 


1-7029 


1.4586 


4.70 


1.5476 


5.1" 


1-6292 


5-50 


1-7047 


1.4609 


4.71 


1.5497 


511 


.-6312 


5-51 


1-7066 


1-4633 


4-72 


1.5518 


5.12 


.-6332 


5-52 


1-7084 


1.4656 


4-73 


1 5539 


513 


1-635. 


S-53 


I-7I02 


1.4679 


4-74 


1.5560 


5.14 


1637. 


5-54 


1-7.20 


1.4702 


4-75 


1.55s. 


5.15 


1-6390 


5-55 


.-7.38 


1-4725 


4.76 


1.5602 


5.16 


1-6409 


5-s6 


1-7156 


'.4748 


4.77 


1.5623 


517 


,6429 


5-57 


1-7174 


1.4770 


4.78 


1.5644 


5.18 


1-6448 


5.5s 


1-7192 


1-4793 


4-79 


1.5665 


5.19 


,-6467 


5-59 


1-7210 


1.48.6 


4.80 


1.5686 


5.20 


.-6487 


5-60 


1-7228 



HYPERBOLIC LOGARITHMS OF NUMBERS. 



63 



i Number. Logarithm. 


Number. 

6.01 
6.02 
6.03 
6.04 
6.05 


Logarithm. 


Number. 


Logarithm. 


Number. 


Logarithm. 


5.61 1.7246 

5.62 1.7263 

5.63 1. 7281 
•5.64 1.7299 

5-65 I 1. 7317 

1 


1.7934 

I-795I 
1.7967 

1.7984 

1. 800 1 


6.41 
6.42 

6.43 
6.44 

6.45 


1.8579 
1.8594 

I.8610 

1.8625 

1. 8641 


6.81 
6.82 
6.83 
6.84 
6.85 


1.9184 
I.9199 
I.9213 
1.9228 
1.9242 


5-66 
5.67 
5.68 

5-69 
5-70 


1-7334 
1-7352 
1.7370 
1-7387 
1.7405 


6.06 
6.07 
6.08 
6.09 
6.10 


1. 8017 
1.8034 
1.8050 
1.8066 
1.8083 


i 6.46 
6.47 
6.48 
6.49 
6.50 


1.8656 
1.8672 
1.8687 
1.8703 
I.8718 


6.86 
6.87 
 6.88 
6.89 
6.90 


1.9257 
1.9272 

1.9286 
I.93OI 
1.9315 


5-71 
5-72 
5-73 
5-74 

1 5-75 

1 


1.7422 
1.7440 

1.7457 

1.7475 
1.7492 


6.U 
6.12 

6.13 
6.14 

6.15 

1 


1.8099 
1.8116 
1.8132 
1.8148 
1.8165 


6.51 
6.52 

6.53 
' 6.54 

: 6.55 


1.8733 
1.8749 

1.8764 

1.8779 

1.8795 


6.91 
6.92 
6.93 
i 6.94 
6.95 


1.9330 
1.9344 
1.9359 

1-9373 
1-9387 


5-76 

5-77 
5-78 

5-79 
5-80 


1.7509 

1.7527 

1.7544 
I. 7561 

1.7579 


6.16 
6.17 
6.18 
6.19 
6.20 


i.8i8i 
1.8197 
1.8213 
1.8229 
1.8245 


6.56 

6.57 
6.58 

6.59 

• 6.60 

1 


1.8810 
1.8825 
1.8840 
1.8856 
1.8871 


6.96 

! 6.97 
6.98 

6.99 
7.00 


1.9402 

1. 9416 

L 1.9430 

1.9445 
1.9459 


5-81 
5-8* 

5-83 
584 
5-85 


1.7596 
I. 7613 

1.7630 
1.7647 

1 . 7664 


6.21 
6.22 
6.23 
6.24 
6.25 


1.8262 
1.8278 
1.8294 
1.8310 
1.8326 


6.61 
6.62 
6.63 
6.64 
6.65 


1.8886 
1. 8901 
1. 8916 

1.8931 
1.8946 


7.01 
7.02 

7.03 
7.04 
7.05 


1-9473 
1.9488 

1.9502 

1.9516 

1.9530 


5-86 

5-87 
5.88 

5-89 
5.90 


1. 7681 
1.7699 
1. 7716 

1-7733 
1.7750 


6.26 
6.27 
6.28 
6.29 
6.30 

1 


1.8342 
1.8358 

1-8374 
1.8390 
1.8405 


6.66 
6.67 
6.68 
6.69 
6.70 


1. 8961 
1.8976 
1.8991 
1.9006 
1. 9021 


7.06 

7.07 
7.08 

7.09 
7.10 


1.9544 

1.9559 

1.9573 

1.9587 
1. 9601 


5-91 
5-92 
5-93 
5-94 
5-95 


1.7766 

1.7783 
1.7800 

1. 7817 

1.7834 1 

1 


6.31 
6.32 

6.33 
6.34 
6.35 


1. 8421 

1.8437 

1.8453 
1.8469 

1.8485 


' 6.71 
6.72 

6.73 
6.74 

6.75 


1.9036 
1.9051 
1.9066 
1. 9081 

1.9095 


7.II 
7.12 

7.13 
7.14 
7.15 


1.9615 
1.9629 
1.9643 

1.9657 
1.9671 


5-96 

5-97 
5.98 

5-99 
. 6.00 

1 
• 


1.7851 
1.7867 
1.7884 
1.7901 
1. 7918 


6.36 

6.37 
6.38 

6.39 
6.40 


1.8500 
1.8516 

1.8532 

1.8547 
1.8563 


6.76 

6.77 

. 6.78 

1 6.79 
6.80 


1.9110 
1.9125 
1. 9140 

1.9155 
1. 9169 


7.16 
7.17 
7.18 
7.19 
7.20 


1.9685 
1.9699 

1.9713 
1.9727 

1. 9741 



MATHEMATICAL TABLES. 





I^,g«i*m. 


Number. 


^.^. 


Number. 


Logarilhn.. 


»„,»,. 


L.«uithin. 




I-97SS 

1.9769 

..9782 


7,61 
7.62 

7-63 
7-64 

7-65 


2.0295 
2.0308 

2.0321 
Z0334 
2-0347 


8.01 
8.02 

8.03 
8.04 
8.05 


2.0807 
2.0819 
2.0832 

2.0844 
2.0857 


8.41 
8.42 
8.43 
8.44 

8-45 


2.1294 

2.1330 
2.1342 




1.9824 
1.9838 
1.9851 
1.986s 
1.9879 


7.66 
7.67 
7.68 
7.69 

7.70 


2.0360 
2-0373 
2.0386 

2.0399 
2.0412 


8.06 
8.07 
8.08 
8.09 
8.10 


2.0869 
2.0882 
2.0894 
2.0906 
2.0919 


S.46 
8.47 
8.48 

If. 


21353 
2.1365 
2.1377 
2.1389 
2.1401 




■.9892 

1.9906 
1.9920 
1.9933 

1-9947 


7.71 
7.72 
7-73 

7.74 
7-75 


2-0425 
2-0438 
2.0451 
2.0464 
2.0477 


8.1 1 

8.12 

8.13 
8.14 
8.15 


2.0931 
2.0943 

2.0980 


8.51 
8.52 
8.53 
8.54 
8.SS 


2.1412 
2.1424 
2.1436 
2.1448 
2.1459 




,.996, 
■9974 
..9988 

2.0001 
2.0015 


7.76 
7-77 
7-78 
7-79 
7.80 


2.0490 
2.0503 
2.0516 

2.0528 
2.0541 


8.16 

8.17 
8.18 
8.19 

8.30 


2.0992 
2.I00S 
2.1017 
2.1029 
2.IO4I 


8.56 
8.57 
8.58 
8.59 
8.60 


2.1471 
2.1483 
2.1494 
2.1506 
2.1518 




2.0028 
2.0042 

2-005 S 
2.0069 
2.0082 


7.81 
7.82 
7-83 
7-84 
7-8s 


2.0554 
2.0567 
2.0580 

2.0605 


8.21 

8.22 

8.23 
8.24 
8.25 


2.1054 
2,1066 
2.1078 
2. 1090 


8.61 
8.62 
8.63 
8.64 
8.65 


2.1529 
2.1541 
2.1552 
2.1564 
2.15,6 




2.0096 
2.0109 

2.0149 


7.86 

7-87 
7.88 
7.89 
7.90 


2.0618 
2.0631 
2.0643 
2.0656 
2.0669 


8.26 
8.27' 
8.28 
8.29 
8.30 


2!lI26 
2.1138 
2. I 150 
2.I163 


8.66 
8.67 
8.68 
8.69 
8.70 


2.1587' 
2.1599 
2.1610 
2.1622 
2.1633 




2.0162 
2.0176 

2'0202 
2.0215 


7.91 

7.92 
7-93 
7-94 
7-95 


2.0681 

2.0694 
2.0707 
2.0719 

2.0732 


8.31 
8.32 
8.33 
8.34 
8-3S 


2.H99 

2.12II 
2.1223 


8.71 
8.72 
8.73 
8.74 
8.75 


2.1645 
2.1656 
2.1668 
2.1679 
2.169, 




2.0229 
2.0242 
2-02SS 
2.0268 
2.0281 


7.96 
7.97 
7.98 
7-99 

8.00 


2.0744 
2.0757 
2.0769 
2.0782 
2-0794 


8.36 
8.37 
8.38 
8.39 
8.40 


2.1235 
2.1247 
2.1258 
2.1270 
2.1282 


8.,6 
8-77 
8.78 
8.79 
8.80 


2.1702 
2.1713 
2.1725 
2.1,36 
2.1748 



HYPERBOLIC LOGARITHMS OF NUMBERS. 



65 



t 

Nanber. 


Logarithm. 


Number. 


Logarithm. 


Number. 


Logarithm. 


Number. 


Logarithm. 


8.81 


2.1759 


9.II 


2.2094 


9.41 


2.2418 


9.71 


2.2732 


8.82 


2.1770 


9.12 


2.2105 


9.42 


2.2428 


9.72 


2.2742 


8.83 


2.1782 


9-13 


2.2116 


9.43 


2.2439 


9.73 


2.2752 


8.84 


2.1793 


9.14 


2.2127 


9-44 


2.2450 


9.74 


2.2762 


8.85 


2.1804 


915 


2.2138 


9.45 


2.2460 


9.75 


2.2773 


8.86 


2.1815 


9.16 


2.2148 


9.46 


2.2471 


9.76 


2.2783 


8.87 


2.1827 


9.17 


2.2159 


9.47 


2.2481 


9-77 


2.2793 


8.88 


2.1838 


9.18 


•2.2170 


9.48 


2.2492 


9.78 


2.2803 


8.89 


2.1849 


9.19 


2.2181 


9.49 


2.2502 


9.79 


2.2814 


8.90 


2.1861 


9.20 


2.219^ 


9.50 


2.2513 


9.80 


2.2824 


8.91 


2.1872 


9.21 


2.2203 


9.51 


2.2523 


9.81 


2.2834 


8.92 


2.1883 


9.22 


2.2214 


9.52 


2.2534 


9.82 


2.2844 


8-93 


2. 1 894 


923 


2.2225 


9.53 


2.2544 


9.83 


2.2854 


8.94 


2. 1 905 


9.24 


2.2235 


9.54 


2.2555 


9.84 


2.2865 


8.95 


2.1917 


9.25 


2.2246 


, 9.55 


2.2565 


9.85 


2.2875 


8.96 


2.1928 


9.26 


2.2257 


9.56 


2.2576 


9.86 


2.2885 


8.97 


2.1939 


9.27 


2.2268 


9.57 


2.2586 


9.87 


2.2895 


8.98 


2.1950 


9.28 


2.2279 


9.58 


2.2597 


9.88 


2.2905 


8.99 


2.1961 


9.29 


2.2289 


9-59 


2.2607 


9.89 


2.2915 


9.00 


2.1972 


9-30 


2.2300 


' 9.60 


2.2618 


9.90 


2.2925 


9.01 


2.1983 


9-31 


2.23II 


9.61 


2.2628 


9.91 


2.2935 


9.02 


2.1994 


932 


2.2322 


9.62 


2.2638 


9.92 


2.2946 


903 


2.2006 


9-33 


2.2332 


9.63 


2.2649 


9-93 


2.2956 


9.04 


2.2017 


9-34 


2.2343 


9.64 


2.2659 


9.94 


2.2966 


9.05 


2.2028 


9-35 


2.2354 


9.65 


2.2670 


9-95 


2.2976 


9.06 


2.2039 


9-36 


2.2364 


i 9.66 


2.2680 


9.96 


2.2986 


9.07 


2.2050 


9-37 


2.2375 


i 967 


2.2690 


9-97 


2.2996 


9.08 


2.2061 


938 


2.2386 


9.68 


2.2701 


9.98 


2.3006 


: 9.09 


2.2072 


9-39 


2.2396 


9.69 


2.271I 


9.99 


2.3016 


9.10 

1 


2.2083 


9.40 


2.2407 


9.70 


2.2721 


10.00 


2.3026 


10.25 


2.3279 


12.75 


2.5455 


15.50 


2.7408 


1 

j 21.0 


3.0445 


10.50 


2.3513 


13.00 


2.5649 


16.0 


2.7726 


22.0 


3.O9II 


10.75 


2.3749 


13.25 


2.5840 


16.5 


2.8034 


23.0 


3.1355 


11.00 


2.3979 


13-50 


2.6027 


17.0 


2.8332 


24.0 


3.I781 


11.25 


2.4201 


13.75 


2.62II 


17.5 


2.8621 


25.0 


3.2189 


11.50 


2.4430 


14.00 


2.6391 


18.0 


2.8904 


26.0 


3.2581 


"75 


2.4636 


14.25 


2.6567 


18.5 


2.9173 


: 27.0 


3.2958 


12.00 


2.4849 


14.50 


2.6740 


19.0 


2.9444 


28.0 


3.3322 


12.25 


2.5052 


14.75 


2.6913 


19.5 


2.9703 


29.0 


33673 


12.50 


2.5262 


15.00 


2.7081 


20.0 


2.9957 


30.0 


3.4012 



MATHEMATICAL , TABLES. 



[IL— NUMBERS, OR DIAMETERS OF CIRCLES, CIR- 
ENCES, AREAS, SQUARES, CUBES, SQUARE ROOTS, 
BE ROOTS. 



0.7854 

 3- 14 
7.07 

 12-57 
19.63 

. 28.^7 
38.48 

. S0.26 
63.61 

 78.54 
95-03 

.113.09 
132-73 
■•53-93 
173-71 
.aoi.o6 
226.98 
.254.46 
28J.52 
.3 14- 15 
346-36 
,380.13 
415-47 
452.38 
490,87 
.530-02 
57^-55 
615-75 
660.52 
.706.85 
754-76 

855-29 
.907.92 
962.11 
:oi7.87 
.075.21 
13411 



.. 256 
289 I 



1.331 
,.1,728 

2,197 ! 

.. 2,744 
3-375 

,.4,096 
4-913 

.- 5,832 
6,859 

■9:261 



15,62s 
17,576 
19,683 



29,791 
3^,768 
35,937 
39,3°4 
42,875 
46,656 
50,653 
54,872 
59,3 '9 
64,000 
68,921 
74,088 



3-162 
3-316 
3-464 
3.605 
5-741 
3.872 
4.000 
4-123 
4.242 
4-358 
4-472 
4-582 

4-795 
4.898 
5.000 
5-099 
5-196 
5-291 
5-385 
S-477 
5-567 
5.656 
5-744 
5-830 
5.916 
6.000 
6.082 
6.164 
6.244 
6.324 
6.403 
6.480 



NUMBERS, OR DIAMETERS OF CIRCLES, &c. 



67 



i? 



Nttjnber, 


Circum- 


CtroUar 


Sauaie. 


Cube. 


Square 


.Cube 


Diameter. 


ference. 


Area. 


^^«4 t*a«a^«« 




Root. 


Root 


43 


135-oS 


1452.20 


1,849 


79,507 


6.557 


3.503 


44 


138.23 


... 1520.52 


... 1,936 


85,184 


6.633 


3.530 


45 


141.37 


1590.43 


2,025 


91,125 


6.708 


3.556 


46 


144.51 


... 1661.90 


... 2,116 


97,336 


6.782 


3.583 


47 


147.65 


1734.94 


2,209 


103,823 


6.855 


3.608 


48 


150-79 


... 1809.55 


... 2,304 


110,592 


6.928 


3-634 


49 


153.93 


1885.74 


2,401 


117,649 


7.000 


3.659 


50 


157.08 


... 1963.49 


... 2,500 


125,000 


7.071 


3.684 


51 


160.22 


2042.82 


2,6oi 


132,651 


7.I4I 


3.708 


52 


163.36 


... 2123.71 


..: 2,704 


140,608 


7. 211 


3.732 


53 


166.50 


2206.18 


2,809 


148,877 


7.280 


3.756 


54 


169.64 


... 2290.21 


... 2,916 


157,464 


7.348 


3.779 


55 


172.78 


2375.82 


3»025 


166,375 


7.416 


3.802 


56 


17592 


... 2463.09 


... 3*136 


I75»6i6 


7.483 


3.825 


57 


179.07 


2551.75 


3i249 


185,193 


7-549 


3.848 


58 


182.21 


... 2642.08 


... 3*364 


i95,"2 


7.61S 


3.870 


59 


185-35 


2733-97 


3,481 


205,379 


7.681 


3.892 


60 


188.49 


... 2827.43 


... 3,600 


216,000 


7.745 


3.914 


61 


19^-63 


2922.46 


3,721 


226,981 


7.810 


3.936 


62 


194.77 


... 3019.07 


... 3,844 


238,328 


7.874 


3.957 


63 


197.92 


3117.24 


3,969 


250,047 


7.937 


3.979 


64 


201.06 


... 3216.99 


... 4^096 


262,144 


8.000 


4.000 


65 


204.20 


3318.30 


4,225 


274,625 


8.062 


4.020 


66 


207.34 


... 3421.18 


... 4,356 


287,496 


8.124 


4.041 


67 


210.48 


3525.65 


4,489 


300,763 


8.185 


4.061 


68 


213.62 


... 3631.68 


... 4,624 


314,432 


8.246 


4.081 


69 


216.77 


3739.28 


4,761 


328,509 


8.306 


4.IOI 


70 


219.91 


... 3848.45 


... 4,900 


343,000 


8.366 


4. 12 I 


71 


223.05 


3959.19 


5,041 


357,911 


8.426 


4.140 


72 


226.19 


... 4071.50 


... 5,184 


373,248 


8.485 


4.160 


73 


22g.Z2» 


4185.38 


5,329 


389,017 


^.544 


4.179 


. 74 


^S^Al 


... 4300.84 


... 5,476 


405,224 


8.602 


4.198 


1 

75 


235.61 


4417.86 


5,625 


421,875 


8.660 


4.217 


76 


238.76 


... 4536.45 


••• 5,776 


438,976 


8.717 


4.235 


77 


241.90 


4656.62 


5,929 


456,533 


8.744 


4.254 


78 1 


245-04 


... 4778.36 


... 6,084 


474,552 


8.831 


4.272 


79 ' 


248.18 


4901.66 


6,241 


493,039 


8.888 


4.290 


1 W ^ 

80 


251-32 


... 5026.54 


... 6,400 


512,000 


8.944 


4.308 


81 ' 25446 


5153.00 


6,561 


531,441 


9.000 


4.326 


82 257.61 


... 5281.01 


... 6,724 


551,368 


9.055 


4.344 


83 260.75 


5410.59 


6,889 


571,787 


9. 1 10 


4.362 


84 


^es-^9 


... 5541.77 


... 7,056 


592,704 


9.165 


4.379 


85 

%6 


267.03 


5674.50 


7,225 


614,125 


9.219 


4.396 


270-17 


... 5808.80 


... 7,396 


636,056 


9.273 


4.414 


87 


273-3^ 


5944.67 


7,569 


658,503 


9.327 


4.431 


88 


276.46 


... 6082.11 


... 7,744 


681,472 


9.380 


4.447 


89 
90 


279.60 


6221.13 


7,921 


704,969 


9.433 


4.461 


282.74 


... 6361.72 


... 8,100 


729,000 


9.486 


4.481 



" ill 



68 



MATHEMATICAL TABLES. 



Number, 










1 




or 


Circum- 


Circular 


Sfluare. 


Cube. 


Square 


Cube 


Diameter. 


ference. 


Area. 




^^%»A#^»« 


Root. 


Root. 


91 


285.88 


6503.87 


8,281 


753*571 


9.539 


4.497 


92 


289.02 


... 6647.61 


... 8,464 


778,688 


9591 


4.514 


93 


292.16 


6792.90 


8,649 


804,357 


9.643 


4.530 


94 


295-31 


... 6939.78 


... 8,836 


830,584 


9.695 


4.546 


95 


298.45 


7088.21 


9>o25 


857*375 


9.746 


4.562 


96 


301-59 


... 7238.23 


... 9,216 


884,736 


9.797 


4.578 


97 


304.73 


7389.81 


9,409 


912,673 


9.848 


4.594 


98 


307.87 


... 7542.96 


... 9,604 


941,192 


9.899 


4.610 


99 


3II.OI 


7697.68 


9,801 


970,299 


9.949 


4.626 


100 


314.15 


-. 7853.97 


...10,000 


... 1,000,000 


10.000 


4.641 


lOI 


317.30 


8011.86 


10,201 


1*030,301 


10.049 


4.657 


102 


320.41 


... 8171.30 


...10,404 


... 1,061,208 


10.099 


4.672 


103 


323.58 


8332.30 


10,609 


1,092,727 


10.148 


4.687 


104 


326.72 


... 8494.88 


...10,816 


.,.. 1,124,864 


10.198 


4.702 


105 


329.86 


8659.03 


11,025 


1*157,625 


10.246 


4.717 


106 


333.00 


... 8824.75 


!... 11,236 


... 1,191,016 


10.295 


4.732 


107 


336.15 


8992.04 


11,449 


1,225,043 


10.344 


4.747 


108 


339.29 


... 9160.90 


...11,664 


... 1,259,712 


10.392 


4.762 


109 


342.43 


9331.33 


11,881 


1,295,029 


10.440 


4.776 


no 


345-57 


- 9503.34 


...12,100 


■•• 1*331,000 


10.488 


4.791 


III 


348.71 


9676.91 


12,321 


1,367.631 


10.535 


4.805 


112 


351.85 


... 9852.05 


...12,544 


... 1,404,928 


10.583 


4.820 


113 


355.01 


10028.77 


12,769 


1,442,897 


10.630 


4.834 


114 


358.14 


...10207.05 


...12,996 


••• 1,481,544 


10.677 


4.848 


115 


361.28 


10386.91 


13*225 


1,520,875 


10.723 


4.862 


116 


364.42 


...10568.34 


... 13*456 


... 1,560,896 


10.770 


4.876 


117 


367.56 


10751.34 


13,689 


1,601,613 


10.816 


4.890 


118 


370.70 


...10935.90 


...13,924 


••• 1,643,032 


10.862 


4.904 


119 


373.81 


III22.O4 


14,161 


1*685,159 


10.908 


4.918 


120 


376.99 


...11309.76 


...14,400 


... 1,728,000 


10.954 


4.932 


121 


380.1^ 


11499.04 


14,641 


1,771,561 


11.000 


4.946 


122 


383.27 


...11689.89 


...14,884 


... 1,815,848 


11.045 


4.959 


123 


386.41 


I1882.3I 


15*129 


1,860,867 


11.090 


4.973 


124 


389.55 


...12076.31 


...15*376 


... 1,906,624 


11.135 


4.986 


125 


392.70 


12271.87 


15*625 


1*953,125 


II. 180 


5.000 


126 


395.84 


...12469.01 


...15*876 


... 2,000,376 


11.224 


5013 


127 


398.98 


12667.71 


16,129 


2,048,383 


11.269 


5.026 


128 


402.12 


...12867.99 


...16,384 


... 2,097,152 


11.313 


5.039 


129 


405.26 


13069.84 


16,641 


2,146,689 


11.357 


5.052 


130 


408.10 


...13273.26 


...16,900 


... 2,197,000 


1 1. 401 


5.065 


131 


411.54 


13478.24 


17,161 


2,248,091 


11.445 


5.078 


132 


414.69 


...13694.80 


...17,424 


... 2,299,968 


11.489 


5.091 


133 


417.83 


13892.94 


17,689 


2,352,637 


".532 


5.104 


134 


420.97 


...14102.64 


...17,956 


... 2,406,104 


.11.575 


5."7 


135 


424.11 


I4313.9I 


18,225 


2,460,375 


II. 618 


5.129 


136 


427.25 


...14526.75 


...18,496 


... 2,515,456 


II. 661 


5.142 


137 


430.39 


I474I.I7 


18,769 


2*571,353 


11.704 


5-155 


138 

1 


433.54 


••.14957.15 


...19,044 


... 2,620,872 


11.747 


5.167 



NUMBERS, OR DIAMETERS OF CIRCLES, ftc. 



69 



Nanaber, 
nr 


Grcnm- 


Circular 


Square. 


Cube. 


Square 


Cube 


EKaxneter. 


f«rence. . 


Area. 






Root. 


Root. 


139 


436.68 


I5174.7I 


19,321 


2,685,619 


11.789 


5.180 


140 


439.82 


•••15393.84 


...19,600 


... 2,744,000 


11.832 


5-192 


141 


442.96 


15614-53 


19,881 


2,803,221 


11.874 


5.204 


142 


446.10 


...15836.80 


...20,164 


... 2,863,288 


11.916 


5-217 


143 


449.24 


16060.64 


20,449 


2,924,207 


11.958 


5.229 


144 


452.39 


...16286.05 


...20,736 


••• 2,985,984 


12.000 


5-241 


145 


455.53 


16513.03 


21,025 


3,048,625 


12.041 


5-253 


146 


45».67 


...16741.58 


...21,316 


... 3,112,136 


12.083 


5.265 


147 


461.81 


16971.70 


21,609 


3,176,523 


12.124 


5-277 


148 


464-95 


...17203.40 


...21,904 


••• 3,241,792 


12.165 


5.289 


149 


468.09 


17436.66 


22,201 


3,307,949 


12.206 


5.301 


150 


471.24 


...17671.50 


...22,500 


••• 3,375,000 


12.247 


5.313 


151 


474.3S 


17907.90 


22,8oi 


3,442,951 


12.288 


5.325 


152 


477.52 


...18145.88 


...23,104 


... 3,511,808 


12.328 


5-336 


153 


480.66 


18385.42 


23,409 


3,581,577 


12.369 


5.348 


154 


483.80 


...18626.54 


...23,716 


... 3,652,264 


12.409 


5.360 


155 , 


486.94 


18869.23 


24,025 


3,723,875 


12.449 


5.371 


156 


490.08 


...19113.49 


••.24,336 


... 3,796,416 


12.489 


5.383 


157 


493.23 


1935932 


24,649 


3,869,893 


12.529 


5.394 


158 


496.37 


...19606.72 


...24,964 


... 3,944,312 


12.569 


5.406 


159 


499.51 


19855.69 


25,281 


4,019,679 


12.609 


5.417 


160 


502.65 


...20106.24 


...25,600 


... 4,096,000 


12.649 


5.428 


161 


505-79 


20358.35 


25,921 


4,173,281 


12.688 


5-440 


162 


508.93 


...20612.03 


...26,244 


... 4,251,528 


12.727 


5.451 


163 


512.08 


20867.20 


26,569 


4,330,747 


12.767 


5.462 


. 164 


515.22 


...21 124. 1 1 


...26,896 


... 4,410,944 


12.806 


5-473 


165 


518.36 


21382.51 


27,225 


4,492,125 


12.845 


S-484 


166 


521.50 


...21642.48 


...27,556 


••• 4,574,296 


12.884 


5.495 


167 


524-64 


21904.02 


27,889 


4,657,463 


12.922 


5-506 


168 


527.78 


...22167.12 


...28,224 


••• 4,741,632 


^12.961 


5-517 


169 


530-93 


22431.80 


28,561 


4,826,809 


13.000 


5.528 


170 1 


534-07 


...22698.06 


...28,900 


••• 4,913,000 


13.038 


5.539 


171 


537.31 


22965.88 


29,241 


5,000,211 


13.076 


5.550 


172 


540.35 


•. 2323527 


...29,584 


... 5,088,448 


13.114 


5.561 


173 


543.49 


23506.23 


29,929 


5,177,717 


13.152 


5.572 


174 


546.03 


...23778.77 


...30,276 


... 5,268,024 


13.190 


5.582 


175 


549. 78 


. 24052.87 


30,625 


5,359,375 


13.228 


5.593 


176 1 


552.92 


...24328.55 


...30,976 


••• 5,451,776 


13.266 


5.604 


177 


556.06 


24605.79 


31,329 


5,545,233 


13.304 


5.614 


178 


559-20 


...24884.61 


...31,684 


- 5,639,752 


13-341 


5.625 


179 


562-34 


25165.00 


32,041 


5,735,339 


13-379 


5.635 


zSo 


565.48 


...25446,96 


...32,400 


... 5,832,000 


13.416 


5.646 


idi 


568.62 


25730.48 


32,761 


5,929,741 


13453 


5.656 


182 


571.77 


...26015.58 


..•33,124 


... 6,028,568 


13.490 


5.667 


183 ; 


574.91 


26302.26 


33,489 


6,128,487 


13^527 


5.677 


184 


573.05 


...26590.50 


...33,856 


... 6,229,504 


13-564 


5.687 


. 1S5 


581.19 


26880.31 


34,225 


6,331,625 


13.601 


5.698 


186 


584.33 


...27171.69 


••34,596 


... 6,434,856 


13.638 


5.708 



70 



MATHEMATICAL TABLES. 



Number, 
or 


Circum- 


Cirailar 


Square. 


Cube. 


Square 


Cube 


Diameter. 


ference. 


Area. 






Root. 


Root. 


187 


537.47 


27464.65 


34,969 


6,539,203. 


13.674 


5.718 


188 


590.62 


..•27759-17 


•-35>344 


... 6,644,672 


I3.71I 


5.728 


189 


593-76 


28055.27 


35,721 


6,751,269 


13-747 


5.738 


190 


596.90 


...28352.94 


...36,100 


... 6,859,000 


13.784 


5.748 


191 


600.04 


28652.17 


36,481 


6,967,871 


13.820 


5.758 


192 


603.18 


...28952.98 


...36,864 


... 7,077,888 


13.856 


5.768 


193 


606.32 


29255.36 


37,249 


7,189,057 


13.892 


5.778 


194 


* 609.47 


-29559.31 


...37,636 


... 7,301,384 


13.928 


5.788 


195 


612.61 


29864.83 


38,025 


7,414,875 


13.964 


5.798 


196 


615.75 


...30171.92 


...38,416 


- 7,529,536 


14.000 


5.808 


197 


618.89 


30480.60 


38,809 


7,645,373 


14.035 


5.818 


198 


622.03 


...30790.82 


...39,204 


... 7,762,392 


14.071 


5.828 


199 


625.17 


31102.52 


39,601 


7,880,599 


14.106 


5.838 


200 


628.32 


...31416.00 


...40,000 


... 8,000,000 


14.142 


5.848 


201 


631.46 


31730.94 


40,401 


8,120,601 


14.177 


5.857 


202 


634.60 


...32047.46 


...40,804 


... 8,242,408 


14.212 


5.867 


203 


637.74 


32365.54 


41,209 


8,365,427 


14.247 


5.877 


204 


640.88 


...32685.20 


...41,616 


... 8,489,664 


14.282 


5-886 


205 


644.02 


33006.43 


42,025 


8,615,125 


' 14.317 


5.896 


206 


647.16 


•33329.23 


...42,436 


... 8,741,816 


14.352 


5.905 


207 


650-31 


33653.60 


42,849 


8,869,743 


14-387 


5-915 


208 


653-45 


-33979.54 


...43,264 


... 8,998,912 


14.422 


5-924 


209 


656.59 


34307.05 


43,681 


9,123,329 


14.456 


5.934 


210 


659-73 


...34636.14 


...44,100 


... 9,261,000 


14.491 


5.943 


211 


662.87 


34966.79 


44,521 


9,393,931 


14.525 


5.953 


212 


666.01 


...35299.01 


-••44,944 


... 9,528,128 


14.560 


5.962 


^13 


669.16 


35632.81 


45,369 


9,663,597 


14.594 


5.972 


214 


672.30 


...35968.17 


...45,796 


... 9,800,344 


14.628 


5.981 


215 


675-44 


36305.11 


46,225 


9,938,375 


14.662 


5.990 


216 


678.5S 


...36643.62 


...46,656 


...10,077,696 


14.696 


6.000 


217 


681.71 


36983.70 


47,089 


10,218,313 


14.730 


6.009 


218 


684.86 


••.37325.34 


..•47,524 


...10,360,232 


14.764 


6.018 


219 


688.01 


37668.56 . 


47,961 


10,503,459 


14.798 


6.027 


220 


691.15 


...38013.36 


...48,400 


...10,648,000 


14.832 


6.036 


221 


694.29 


38359-72 


48,841 


10,793,861 


14.866 


6.045 


222 


697.43 


...38707.65 


...49,284 


...10,941,048 


14.899 


6.055 


223 


700.57 


39037.51 


49,729 


11,089,567 


14.933 


6.064 


224 


703-71 


...39408.23 


...50,176 


...11,239,424 


14.966 


6.073 


225 


706.86 


39760.87 


50,625 


11,390,625 


15.000 


6.082 


226 


710.00 


...40115.09 


...51,076 


--•",543,176 


15-033 


6.091 


227 


713.14 


40470.87 


51,529 


11,697,083 


15.066 


6.100 


228 


716.28 


...40828.23 


...51,984 


••.11,852,352 


15.099 


6.109 


229 


719.42 


41187.16 


52,441 


12,008,989 


15-132 


6. 1 18 


230 


722.56 


...41547.66 


...52,900 


...12,167,000 


15-165 


6.126 


231 


725.70 


41909.72 


53,361 


12,326,391 


15.198 


6.135 


232 


728.85 


...42273.36 


-..53,824 


...12,487,168 


15.231 


6.144 


233 


731.99 


• 42638.58 


54,289 


12,649,337 


15.264 


6.^53 


234 


735.13 


...43005.36 


..-54,756 


...12,812,904 


15.297 


6.162 



NUMBERS, OR DIAMETERS OF CIRCLES, &c. 



71 



NnTTiber, 














or 


Circum- 


Circular 


Square. 


Cube. 


Square 


Cube 


Dumeter 


ference. 

• 1 


Area. 






Root. 


Root. 


235 


i 738.27 


43373-71 


55,225 


12,977,875 


15329 


6.171 


236 


741.41 


...43743-63 


...55,696 


...13,144,256 


15.362 


6.179 


237 


744-55 


44115.11 


56,169 


13,312,053 


15.394 


6.188 


23S 


747.68 


...44488.19 


...56,644 


...13,481,272 


15.427 


6.197 


239 


750.88 


44862.83 


57,121 


13,651,919 


15-459 


6.205 


240 


753.98 


...45239.04 


...57,600 


...13,824,000 


I549I 


6.214 


241 


757.12 


45616.81 


58,081 


13,997,521 


15.524 


6.223 


242 


760.26 


...45996.16 


...58,564 


...14,172,488 


15.556 


6.231 


243 


763.40 


46377.08 


59,049 


14,348,907 


15.588 


6.240 


244 


766.52 


...46759.57 


...59,536 


...14,526,784 


15.620 


6.248 


245 


769.92 


47143-63 


60,025 


14,706,125 


15^652 


6.257 


246 


' 772.83 


...47529.26 


...60,516 


...14,886,936 


15.684 


6.265 


247 


775-97 


47916.46 


61,009 


15,069,223 


15.716 


6.274 


248 


779.11 


...48305.24 


...61,504 


...15,252,992 


15.748 


6.282 


249 


782.25 


48695.58 


62,001 


15,438,249 


15.779 


6.291 


250 


785.40 


...49087.50 


...62,500 


...15,625,000 


I5.81I 


6.299 


251 


788.54 


49480.98 


63,001 


15,813,251 


15.842 


6.307 


252 


791.68 


...49876.04 


...63,504 


...16,003,008 


15.874 


6.316 


253 


794.82 


50272.66 


64,009 


16,194,277 


15.905 


6.324 


254 


797.96 


...50670.86 


...64,516 


...16,387,064 


15.937 


6.333 


255 


801.10 


51070.63 


65,025 


16,581,375 


15.968 


6.341 


256 


804.24 


...51471-96 


...65,536 


...16,777,216 


16.000 


6.349 


257 


807.39 


51874.88 


66,049 


16,974,593 


16.031 


6.357 


258 


810.53 


...52279.36 


...66,564 


••.17,173,512 


16.062 


6.366 


259 


813.67 


52685.41 


67,081 


17,373,979 


16.093 


6.374 


2€o 


816.81 


...53093.04 


...67,600 


•..17,576,000 


16.124 


6.382 


, 261 


819.95 


53502.23 


68,121 


17,779,581 


16.155 


6.390 


j 262 


823.09 


...53912.99 


...68,644 


...17,984,728 


16.186 


6.398 


263 


826.24 


54325.33 


69,169 


18,191,447 


16.217 


6.406 


264 


829.38 


-54739.23 


...69,696 


•••18,399,744 


16.248 


6.415 


265 


832.52 


55154.71 


70,225 


18,609,625 


f6.278 


6.423 


266 


835.66 


...55571.76 


...70,756 


...18,821,096 


16.309 


6.431 


267 


838.80 


55990.38 


71,289 


19,034,163 


16.340 


6.439 


268 


841.94 


...56410.56 


...71,824 


...19,248,832 


16.370 


6.447 


269 


845-09 


56832.32 


72,361 


19,465,109 


16.401 


6.455 


270 


848.23 


...57255.66 


...72,900 


...19,683,000 


16.431 


6.463 


271 


851.37 


57680.56 


73,441 


19,902,511 


16.462 


6.471 


272 


854-51 


...58107.03 


...73,984 


...20,123,648 


16.492 


6.479 


. ^^^ 


857.65 


58535.07 


74,529 


20,346,417 


16.522 


6.487 


1 274 


860.79 


...58964.69 


...75,076 


...20,570,824 


16.552 


6.495 


275 


863.94 


59393.87 


75,625 


20,796,875 


16.583 


6.502 


276 ; 


867.08 


...59828.63 


...76,176 


...21,024,576 


16.613 


6.510 


277 


870.22 


60262.95 


76,729 


21,253,933 


16.643 


6.518 


278 


873.36 


...60698.85 


...77,284 


...21,484,952 


16.673 


6.526 


279 


876.50 


61136.32 


77,841 


21,717,639 


16.703 


6.534 


280 


879.64 


...61573.36 


...78,400 


...21,952,000 


16.733 


6.542 


281 


882.78 


62015.96 


78,961 


22,188,041 


16.763 


6.549 


282 


885.93 


...62458.14 


...79,524 


...22,425,768 


16.792 


6.557 



/ 



/ 



/ 



\ 



72 



MATHEMATICAL TABLES. 



Number. 














A « «• AAA l^^^s J 

or 


Circum- 


Circular 


Square. 


Cube. 


Square 


Cube 


Diameter. 


ference. 


Area. 






Root. 


Root. 


283 


889.07 


62901.90 


80,089 


22,665,187 


16.822 


6.565 


284 


892.21 


...63347.22 


...80,656 


...22,906,304 


16.852 


6.573 


285 


895-35 


63794.11 


81,225 


23,149,125 


16.881 


6.580 


286 


898.49 


...64242.57 


...81,796 


•••23,393,656 


16.9II 


6.588 


287 


901.63 


64692.61 


82,369 


23,639,903 


16.941 


6.596 


288 


904.78 


...65144.21 


...82,944 


...23,887,872 


16.970 


6.603 


289 


907.92 


65597-39 


83,521 


24,137,569 


17.000 


6.611 


290 


911.06 


...66052.14 


...84,100 


...24,389,000 


17.029 


6.619 


291 


914.20 


66508.45 


84,681 


24,642,171 


17.059 


6.627 


292 


91734 


...66966.34 


...85,264 


...24,897,088 


17.088 


6.634 


293 


920.48 


67425.80 


85,849 


25*153,757 


I7.II7 


6.642 


294 


923.63 


...67886.83 


...86,436 


...25,412,184 


17.146 


6.649 


295 


926.77 


68349.43 


87,025 


25*672,375 


17.176 


6.657 


296 


929.91 


...68813.60 


...87,616 


...25,934,336 


17.205 


6.664 


297 


933-05 


69279.34 


88,209 


26,198,073 


17.234 


6.672 


298 


936.19 


...69746.66 


...88,804 


...26,463,592 


17.263 


6.679 


299 


939-33 


. 70215.54 


89,401 


26,730,899 


17.292 


6.687 


300 


942.48 


...70686.00 


...90,000 


...27,000,000 


17.320 


6.694 


301 


945.62 


71158.02 


90,601 


27,270,901 


17.349 


6.702 


302 


948.76 


...71631.62 


...91,204 


...27,543,608 


17.378 


6.709 


303 


951-90 


72106.78 


91,809 


27,818,127 


17.407 


6.717 


304 


955-04 


-••72583.52 


...92,416 


...28,094,464 


17.436 


6.724 


305 


958.18 


73061.83 


93,025 


28,372,625 


17.464 


6.731 


306 


961.32 


..•73541.71 


•93,636 


...28,652,616 


17.493 


6.739 


307 


964.47 


74023.16 


94,249 


28,934,443 


^7.521 


6.746 


308 


967.61 


...74506.18 


...94,864 


...29,218,112 


17.549 


6.753 


309 


970.75 


74990.77 


95,481 


29,503,629 


17-578 


6.761 


310 


973-89 


...75476.94 


...96,100 


...29,791,000 


17.607 


6.768 


311 


977-03 


75964.67 


96,721 


30,080,231 


17.635 


6.775 


312 


980. 1 7 


...76453.93 


...97,344 


.. 30,371*328 


17.663 


6.782 


313 


983.32 


76944-85 


97,969 


30,664,297 


17.692 


6.789 


314 


986.45 


...77437.29 


...98,596 


...30,959,144 


17.720 


6.797 


315 


989.60 


77931.31 


99,225 


31,255,875 


17.748 


6.804 


316 


992.74 


...78426.89 


...99,856 


.. .31*554,496 


17.776 


6.81 1 


317 


995-88 


78924.06 


100,489 


31*855,013 


17.804 


6.818 


318 


999.02 


...79422.78 


101,124 


•..32,157,432 


17.832 


6.826 


319 


1002.17 


79923.08 


101,761 


32,461,759 


17.860 


6.833 


320 


1005.31 


...80424.96 


102,400 


...32,768,000 


17.888 


6.839 


321 


1008.45 


80928.40 


103,041 


33,076,161 


17.916 


6.847 


322 


1011.59 


...81433.41 


103,684 


...33*386,248 


17-944 


6.854 


323 


1014.73 


81939.99 


104,329 


33,698,267 


17.97^2 


6.861 


324 


1017.47 


...82448.15 


104,976 


...34,012,224 


18.000 


6.868 


325 


1021.02 


82957.87 


105,625 


34,328,125 


18.028 


6.875 


326 


1024.16 


...83469.17 


106,276 


..•34,645*976 


18.055 


6.882 


327 


1027.30 


83982.60 


106,929 


34,965*783 


18.083 


6.889 


328 


1030.44 


...84496.47 


107,584 


...35*287,552 


18.IH 


6.896 


329 


1033.58 


85012.48 


108,241 


35,611,289 


18.138 


6.903 


330 


1036.72 


...85530.06 


108,900 


•-.35,937*000 


18.166 


6.910 



NUMBERS, OR DIAMETERS OF CIRCLES, Ac. 



73 





Nomber, 
















or 


Grcum- 


Circular 


Square. 


Cube. 


Square 


Cube 




Diameter. 


ference. 


Area. 






Root. 


Root. 


331 


1039.86 


86049.20 


109,561 


36,264,691 


18.193 


6.917 




332 


1043.01 


...86569.92 


110,224 


...36,594,368 


18.221 


6.924 




333 


1046.15 


87092.22 


110,889 


36,926,037 


18.248 


6.931 




334 


1049.29 


...87616.08 


111,556 


..•37,259,704 


18.276 


6.938 




335 


1052.43 


88141.51 


112,225 


37,595,375 


18.303 


6.945 




336 


1055.57 


...88668.51 


112,896 


...37^33,056 


18.330 


6.952 




337 


1058.71 


89197.09 


113,569 


38,272,753 


18.357 


6.959 




338 


1061.86 


...89727.23 


114,244 


...38,614,472 


18.385 


6.966 




339 


1065.02 


90258.9s 


114,921 


38,958,219 


18.412 


6.973 




340 


1068.14 


...90792.24 


115,600 


...39,304,000 


18.439 


6.979 




341 


1071.28 


91327.09 


116,281 


39,651,821 


18.466 


6.986 




342 


1074.27 


■..91863.52 


116,964 


...40,001,688 


18.493 


6.993 




343 


1077.56 


92401.15 


117,649 


40,353,607 


18.520 


7.000 




344 


1080.71 


...92941.09 


118,336 


...40,707,584 


18.547 


7.007 




345 


1083.85 


93482.23 


119,025 


41,063,625 


18.574 


7.014 




346 


1086.99 


...94024.94 


119,716 


...41,421,736 


18.601 


7.020 




347 


1090.35 


94569.22 


120,409 


41,781,923 


18.628 


7.027 




348 


1093.07 


...95115.08 


121,104 


...42.144,192 


18.655 


7.034 


349 


1096.41 


95662.50 


I2I,8oi 


42,508,549 


18.681 


7.040 


350 


1099.56 


...96211.50 


122,500 


...42,875,000 


18.708 


7.047 




351 


1102.70 


96762.06 


123,201 


43,243,551 


18.735 


7.054 




352 


1105.84 


...97314.20 


123,904 


...43,614,208 


18.762 


7.061 




353 


1 108.98 


97867.90 


124,609 


43,986,977 


18.788 


7.067 




354 


III2.62 


...98423.18 


125,316 


...44,361,864 


18.815 


7.074 




355 


III5.26 


98980.03 


126,025 


44,738,875 


18.842 


7.081 




356 


1 118.40 


..•99538.45 


126,736 


...45,118,016 


18.868 


7.087 




357 


II2I.55 


100098.43 


127,449 


45,499,293 


18.894 


7.094 




358 


1124.69 


100660.00 


128,164 


...45,882,712 


18.921 


7.101 




359 


1127.83 


101223.13 


128,881 


46,268,279 


18.947 


7.107 




360 


1130.97 


101787.84 


129,600 


...46,656,000 


18.974 


7.114 




361 


II34.II 


102354. H 


130,321 


47,045,881 


19.000 


7.120 




362 


, "37.25 


102921.95 


131,044 


.••47,437,928 


19.026 


7.127 




3^3 


1140.40 


103491.31 


131,769 


47,832,147 


19.052 


7.133 




364 


1143.54 


104062.35 


132,496 


...48,228,544 


19.079 


7.140 




365 


1146.68 


104634.91 


133,225 


48,627,125 


19.105 


7.146 




366 


1 149.82 


105209.04 


133,956 


...49,027,896 


19.131 


7.153 


 367 


1152.96 


10578474 


134,689 


49,430,863 


19.157 


7.159 


368 


1 156.10 


106362.00 


135,424 


•..49,836,032 


19.183 


7.166 




369 


1159.25 


106940.84 


136,161 


50,243,409 


19.209 


7.172 




370 


1162.39 


107521.26 


136,900 


.-.50,653,000 


19.235 


7.179 




371 


1165.53 


108103.22 


137,641 


51,064,811 


19.261 


7.185 


372 


1168.67 


108686.79 


138,384 


...51,478,848 


19.287 


7.192 


373 


II7I.81 


IO9271.91 


139,129 


51,895,117 


19.313 


7.198 


t 


374 


1 17495 


109858.62 


139,876 


...52,313,624 


19.339 


7.205 




375 


II78.IO 


110446.87 


140,625 


52,734,375 


19.365 


7. 211 




376 


1 181.24 


IIIO36.71 


141,376 


...53,157,376 


19391 


7.218 




377 


1 18438 


III628.II 


142,129 


53,582,633 


19.416 


7.224 




378 


1187.52 


II222I.O9 


142,884 


...54,010,152 


19.442 


7-230 



74 



MATHEMATICAL TABLES. 



i Number, 
or 


Circum-b 


Circular 


Square. 


Cube. 


Square ( 


iTube 


Diameter. 


ference. 


Arcx 






Root. I 


looL 


379 


1190.66 


II2815.64 


143,641 


54,439,939 


19.468 7. 


237 


380 


1193.80 


II34II.76 


144,400 


...54,872,000 


19.493 7 


•243 


381 


1196.94 


114009.46 


145,161 


55,306,341 


19.519 7. 


249 


382 


1200.09 


114608.70 


145*924 


•..55,742,968 


19-545 7' 


256 


383 


1203.23 


115209.54 


146,689 


56,181,887 


19.570 7. 


,262 


384 


1206.37 


115811*94 


147,456 


...56,623,104 


19.596 7. 


268 


385 


1209.51 


I16415.9I 


148,225 


57,066,625 


19.621 7. 


'275 


386 


1212.65 


117021.45 


148,996 


...57,512,456 


19.647 7. 


.281 


387 


1215.79 


117628.57 


149,769 


57,960,603 


19.672 7 


.287 


388 


1218.94 


118237.25 


150,544 


...58,411,072 


19.698 7. 


.294 


389 


1222.08 


I18846.51 


151,321 


58,863,869 


19.723 7. 


299 


390 


1225.22 


119453.94 


152,100 


-••59,319,000 


19.748 7 


■306 


391 


1228.36 


120072.73 


152,881 


59,776,471 


19.774 7. 


312 


392 


1231.50 


120687.70 


153,664 


...60,236,288 


19.799 7< 


319 


393 


1234.64 


121304.24 


154,449 


60,698,457 


19.824 7, 


.325 


394 


1237.79 


121922.43 


155,236 


...61,162,984 


19.849 7. 


331 


395 


1240.93 


122542.03 


156,025 


61,629,875 


19-875 7- 


337 


396 


1244.07 


123163.28 


156,816 


...62,099,136 


19.899 7 


343 


397 


1247.21 


123786.10 


157,609 


62,570,773 


19.925 7 


■349 


398 


1250.35 


I24412.IO 


158,404 


•..63,044,792 


19.949 7 


.356 


399 


1253.49 


125036.46 


159,201 


63,521,199 


19.975 7 


362 


400 


1256.64 


125664.00 


160,000 


...64,000,000 


20.000 7 


■368 


401 


1259.78 


126293.10 


160,801 


64,481,201 


20.025 7. 


.374 


402 


1262.92 


126923.88 


161,604 


...64,964,808 


20.049 7 


.380 


403 


1266.06 


127556.02 


162,409 


65,450,827 


20.075 7. 


.386 


404 


1269.20 


128189.84 


163,216 


••65,939,264 


20.099 7. 


•392 


405 


1272.34 


128825.23 


164,025 


66,430,125 


20.125 7 


•399 


406 


1275.48 


129462.19 


164,836 


...66,923,416 


20.149 7. 


405 


407 


1278.63 


I30IOO.71 


165,649 


67,419,143 


20.174 7. 


.411 


408 


1281.77 


130740.82 


166,464 


...67,911,312 


20.199 7 


.417 


409 


1284.91 


131382.49 


167,281 


68,417,929 


1 20.224 7 


.422 


410 


1288.05 


132025.74 


168,100 


...68,921,000 


20.248 7. 


.429 


411 


I29I.19 


132670.55 


168,921 


69,426,531 


20.273 7. 


434 


412 


1294.32 


133316.93 


169,744 


••-69,934,528 


20.298 7. 


441 


413 


1297.48 


133964.89 


170,569 


70,444,997 


20.322 7 


■447 


414 


1300.62 


134614.41 


171,396 


-■•70,957,944 


20.347 7 


■453 


415 


1303-76 


135265.51 


172,225 


71,473,375 


20.371 7. 


■459 


416 


1306.90 


I35918.18 


173,056 


...71,991,296 


20.396 7. 


465 


417 


1310.04 


136572.42 


173,889 


72,511,713 


20.421 7. 


.471 


418 


I313.18 


137228.22 


174,724 


• -73,034,632 


20.445 7- 


477 


419 


1316.32 


137885.69 


175,561 


73,560,059 


20.469 7. 


.483 


420 


1319-47 


138544.56 


176,400 


...74,088,000 


20.494 7. 


489 


421 


1322.61 


139205.08 


177,241 


74,618,461 


20.518 7 


•495 


422 


1325.75 


139867.17 


178,084 


...75,151,448 


20.543 7- 


.501 


423 


1328.89 


140530.83 


178,929 


75,686,967 


20.567 7 


■507 


424 


1332.03 


I41196.07 


179,776 


...76,225,024 


20.591 7. 


•513 


425 


1335-18 


141862.87 


180,625 


76,765,625 


20.615 7 


.518 


426 


1338.32 


142531.25 


181,476 


•.■77,308,776 


\ 20.639 7 


■524 



NUMBERS, OR DIAMETERS OF CIRCLES, 4c. 



75 



Nnmber, 
or 


Grcum- 


Circular 


S<]uare. 


Cube. 


Square 


Cube 


Diameter. 


ference. 


Area. 






Root. 


Root. 


427 


1341.46 


I432OI.I9 


182,329 


77,854,483 


20.664 


7.530 


428 


1344.60 


143872.71 


183,184 


...78,402,752 


20.688 


7.536 


429 


1347.74 


144545.80 


184,041 


78,953,589 


20.712 


7.542 


430 


1550.88 


145220.46 


184,900 


...79,507,000 


20.736 


7.548 


431 


1354.02 


145696.68 


185,761 


80,062,991 


20.760 


7.554 


432 


1357.17 


14657448 


186,624 


...80,621,568 


20.785 


7-559 


433 


1360.33 


147253.85 


187,489 


81,182,737 


20.809 


7.565 


434 


1363.45 


147934.80 


188,356 


...81,746,504 


20.833 


7.571 


435 


1366.59 


I48617.3I 


189,225 


. 82,312,875 


20.857 


7.577 


436 


1369.73 


149301.39 


190,096 


...82,881,856 


20.881 


7.583 


437 


1372.87 


149987.05 


190,969 


83,453,453 


20.904 


7.588 


438 


1376.02 


150674.27 


191,844 


...84,027,672 


20.928 


7.594 


439 


1379.16 


151362.87 


192,721 


84,604,519 


20.952 


7.600 


440 


1382.30 


152053.44 


193,600 


...85,184,000 


20.976 


7.606 


441 


1385.44 


152745.37 


194,481 


85,766,121 


21.000 


7.612 


442 


1388.58 


153438.88 


195*364 


.■.86,350,388 


21.024 


7.617 


443 


1391.72 


154133.96 


196,249 


86,938,307 


21.047 


7.623 


444 


1394.87 


154830.61 


197,136 


...87,528,384 


21.071 


7.629 


445 


1398.01 


155528.83 


198,025 


88,121,125 


21.095 


7.635 


446 


I4OI.I5 


156228.62 


198,916 


...88,716,536 


21. 119 


7.640 


447 


1404.29 


156929.98 


199,809 


89,314,623 


21.142 


7.646 


448 


1407.43 


157632.92 


200,704 


•••89,915,392 


21.166 


7.652 


449 


1410.57 


158337.42 


201,601 


90,518,849 


21.189 


7.657 


450 


1413.72 


159043.50 


202,500 


...91,125,000 


21.213 


7.663 


451 


1416.86 


I5975I.I4 


203,401 


91,733,851 


21.237 


7.669 


452 


1420.00 


160460.36 


204,304 


...92,345,408 


21.260 


7.674 


453 


1423.14 


161171.14 


205,209 


92,959,677 


21.284 


7.680 


454 


1426.28 


161883.50 


206,106 


•••93,576,664 


21.307 


7.686 


455 


1429.42 


162597.43 


207,025 


94,196,375 


21.331 


7.691 


456 


1432.56 


163312.93 


207,936 


...94,818,816 


21.354 


7.697 


' 457 


1435.71 


164030.20 


208,849 


95,443,993 


21.377 


7703 


45« 


1438.85 


164748.64 


209,764 


...96,071,912 


21.401 


7.708 


459 


1441.99 


165468.85 


210,681 


96,702,579 


21.424 


7.714 


460 


1445-13 


166190.64 


211,600 


• -97,336,000 


21.447 


7.719 


, 461 


1448.27 


166913.99 


212,521 


97,972,181 


21.471 


7.725 


462 


I45I.4I 


167638.91 


213,444 


...98,611,128 


21.494 


7.731 


463 


1454.56 


168365.41 


214,369 


99,252,847 


21.517 


7.736 


464 


1457.70 


169093.47 


215,296 


•••99,897,345 


21.541 


7.742 


. 465 


1460.84 


169823.II 


216,225 


100,544,625 


21.564 


7.747 


. 466 


1463.98 


17055432 


2171I56 


101,194,696 


21.587 


7.753 


I 467 


1467.12 


I71287.IO 


218,089 


101,847,563 


21.610 


7.758 


1 468 


1470.26 


172021.44 


219,024 


102,503,232 


21.633 


7.764 


! 469 


1473.41 


172757.36 


219,961 


103,161,709 


21.656 


7.769 


470 


1476.55 


173494.86 


220,900 


103,823,000 


21.679 


7^775 


471 


1479.69 


174233.92 


221,841 


104,487,111 


21.702 


7.780 


i ^72 


1482.83 


17497455 


222,784 


105,154,048 


21.725 


7.786 


473 


1485.97 


175716.75 


223,729 


105,823,817 


21.749 


7.791 


474 


1 489. 1 1 


176460.45 


224,676 


166,496,424 


21.771 


7.797 



1^ 



MATHEMATICAL TABLES. 



Number, 
or 


Circum- 


Circular 


Soimrc 


Cube. 


Square 


Cube 


Diameter. 


ference. 


Area. 


fcj*l >M1 i ^* 




Root. 


Root. 


475 


1492.26 


177205.87 


225,625 


107,171,875 


21.794 


7.802 


476 


1495.36 


177952.79 


226,576 


107,850,176 


21.817 


7.808 


477 


1498.54 


178701.27 


227,529 


108,531,333 


21.840 


7.813 


478 


1501.68 


179451.33 


228,484 


109,215,352 


21.863 


7.819 


479 


1504.82 


180202.96 


229,441 


109,902,239 


21.886 


7.824 


480 


1507.96 


180956.16 


230,400 


110,592,000 


21.909 


7.830 


481 


I5II.IO 


181712.92 


231,361 


111,284,641 


21.932 


7.835 


482 


1514.25 


182467.26 


232,324 


111,980,168 


21.954 


7.840 


483 


1517.39 


183225.18 


233,289 


112,678,587 


21.977 


7.846 


484 


1520.53 


183984.66 


234,256 


"3,379,904 


22.000 


7.851 


485 


1523.67 


184745.71 


235,225 


114,084,125 


22.023 


7.857 


486 


1526.81 


185508.33 


236,196 


114,791,256 


22.045 


7.862 


487 


1529.95 


186272.53 


237,169 


115,501,303 


22.069 


7.868 


488 


1533.90 


187038.29 


238,144 


116,214,272 


22.091 


7.873 


489 


1536.24 


187805.63 


239,121 


116,936,169 


22.113 


7.878 


490 


1539.38 


188574,54 


240,100 


117,649,000 


22.136 


7.884 


491 


1542.52 


189345.01 


241,081 


118,370,771 


22.158 


7.889 


492 


1545.66 


I9OII7.06 


242,064 


119,095,488 


22.181 


7.894 


493 


1548.80 


190890.68 


243,049 


119,823,157 


22.204 


7.899 


494 


1551.95 


191665.87 


244,036 


120,553,784 


22.226 


7.905 


495 


155509 


192442.63 


245,025 


121,287,375 


22.248 


7.910 


496 


1558.23 


193220.96 


246,016 


122,023,936 


22.271 


7.915 


497 


1561.37 


194000.86 


247,009 


122,763,473 


22.293 


7.921 


498 


1564.51 


194782.34 


248,004 


123,505,992 


22.316 


7.926 


499 


1567.55 


195565.38 


249,001 


124,251,499 


22.338 


7.932 


500 


1570.80 


196350.00 


250,000 


125,000,000 


22.361 


7.937 


501 


1573.94 


I97136.18 


251,001 


125,751,501 


22.383 


7.942 


502 


1577.08 


197923.94 


252,004 


126,506,008 


22.405 


7.947 


503 


1580.22 


198713.26 


253,009 


127,263,527 


22.428 


7.953 


504 


1583.36 


199504.16 


254,016 


128,024,864 


22.449 


7.958 


505 


1586.50 


200296.63 


255,025 


128,787,625 


22.472 


7-963 


506 


1589.64 


201090.67 


256,036 


129,554,216 


22.494 


7.969 


507 


1592.79 


201886.28 


257,049 


130,323,843 


22.517 


7.974 


508 


1595.93 


202683.46 


258,064 


131,096,512 


22.539 


7.979 


509 


1599.07 


203487.70 


259,081 


131,872,229 


22.561 


7.984 


510 


1602.21 


204282.54 


260,100 


132,651,000 


22.583 


7.989 


5" 


1605.35 


205084.43 


261,121 


133,432,831 


22.605 


7.995 


512 


1608.49 


205887.84 


262,144 


134,217,728 


22.627 


8.000 


513 


161I.64 


206692.93 


263,169 


135,005,697 


22.649 


8.005 


514 


1614.78 


207499.53 


264,196 


135,796,744 


22.671 


8.010 


515 


1617.92 


208307.71 


265,225 


136,590,875 


22.694 


8.016 


516 


1621.06 


209117.46 


266,256 


137,388,096 


22.716 


8.021 


517 


1624.20 


209928.78 


267,289 


138,188,413 


22.738 


8.026 


S18 


1627.34 


210741.66 


268,324 


138,991,832 


22.759 


8.031 


519 


1630.49 


2II556.I2 


269,361 


139,798,359 


22.782 


8.036 


520 


1633.63 


212372.16 


270,400 


140,608,000 


22.803 


8.041 


521 


1636.77 


213189.76 


271,441 


141,420,761 


22.825 


8.047 


522 


1639.93 


214008.93 


272,484 


142,236,648 


22.847 


8.052 



NUMBERS, OR DIAMETERS OF CIRCLES, &c. 



77 



' Nnmber, 














or 


OrcoiB- 


Circular 


Square. 


Cube, 


Square 


Cube 


iDofDClCT. 




Area. 




^0^mm^^0^ 


Root. 


Root. 


S»3 


1643.05 


214829.67 


273,529 


143,055,667 


22.869 


8.057 


5«4 


1646.19 


215651.99 


274,576 


143,877,824 


22.891 


8.062 


5»5 


1649.34 


216475-87 


275,625 


144,703,125 


22.913 


8.067 


526 


1652.48 


217301.33 


276,676 


145,531,576 


22.935 


8.072 


527 


1655.62 


218128.35 


277,729 


146,363,183 


22.956 


8.077 


5*8 


1658.76 


218956.95 


278,784 


147,197,952 


22.978 


8.082 


5*9 


1661.90 


219787.12 


279,841 


148,035,889 


23.000 


8.087 


530 


1665.04 


220618.86 


280,900 


148,877,000 


23.022 


8.093 


531 


1668.18 


221452.16 


281,961 


149,721,291 


23.043 


8.098 


' 53* 


1671-33 


222287.04 


283,024 


150,568,768 


23.065 


8.103 


533 


1674,47 


223123.50 


284,089 


151,419,437 


23-087 


8.108 


534 


* 1677.61 


223961.52 


285,156 


152,273,304 


23.108 


8.II3 


535 


1680.75 


224801. II 


286,225 


153,130,375 


23.130 


8. 118 


536 


1683.80 


225642.27 


287,296 


153,990,656 


23.152 


8.123 


537 


I 1687.04 


226487.01 


288,369 


154,854,153 


23-173 


8.128 


538 


. 1690.18 


2273293^ 


289,444 


155,720,872 


23-195 


8.133 


539 


1 1693.32 


228175.19 


290,521 


156,590,819 


23.216 


8.138 


540 


1696.46 


229022.64 


291,600 


157,464,000 


23-238 


8.143 


541 


1699.60 


229871.65 


292,681 


158,340,421 


23-259 


8.148 


542 


1702.74 


230722.24 


293,764 


159,220,088 


23.281 


8.153 


543 


1705.88 


231574.40 


294,849 


160,103,007 


23.302 


8.158 


544 


1709.03 


232428.13 


295,936 


160,989,184 


23-324 


8.163 


545 


1712.17 


233283.43 


297,025 


161,878,625 


23-345 


8.168 


546 


' 1715-31 


234140.30 


298,116 


162,771,336 


23367 


8.173 


' 547 


1718.45 


234998.74 


299,209 


163,667,323 


23-388 


8.178 


548 


. 1721.59 


235858.76 


300,304 


164,566,592 


23-409 


8.183 


549 


• 1724.73 


236720.34 


301,401 


165,469,149 


23-431 


8.188 


550 


' 1727.88 


237583-50 


302,500 


166,375,000 


23-452 


8.193 


551 


1731.02 


238448.22 


303,601 


167,284,151 


23-473 


8.198 


55* 


1734.16 


239314.52 


304,704 


168,196,608 


23495 


8.203 


' 553 


1737.30 


240182.38 


305,809 


169,112,377 


235^6 


8.208 


554 


I 1740.44 


241051.82 


306,916 


170,031,464 


23-537 


8.213 


555 


1743.58 


241922.83 


308,025 


170,953,875 


23-558 


8.218 


556 


1746.72 


242795.41 


309,136 


171,879,616 


23-579 


8.223 


557 


1749.77 


243669.56 


310,249 


172,808,693 


23.601 


8.228 


558 


1753.09 


244545.28 


311,364 


173,741,112 


23.622 


8.233 


559 


1756.^^5 


245422.57 


312,481 


174,676,879 


23-643 


8.238 


560 


1759.29 


246301.44 


313,600 


175,616,000 


23.664 


8.242 


561 


1762.43 


247181.87 


314,721 


176,558,481 


23.685 


8.247 


562 


! 1765.57 


248063.87 


315,844 


177,504,328 


23.706 


8.252 


563 


1768.72 


248947.45 


316,969 


178,453,547 


23.728 


8.257 


564 


1771.86 


249832.59 


318,096 


179,406,144 


23.749 


8.262 


565 


1775.00 


250719-31 


319,225 


180,362,125 


23.769 


8.267 


566 


1778.14 


251607.60 


320,356 


181,321,496 


23.791 


8.272 


567 


1781.28 


252497.36 


321,489 


182,284,263 


23.812 


8.277 


568 


1 1784.42 


253388.88 


322,624 


183,250,432 


23833 


8.282 


569 


1787.57 


254281.88 


323,761 


184,220,009 


23-854 


8.286 


570 

1 


1790.71 


255176.64 


324,900 


185,193,000 


23-875 


8.291 



78 



MATHEMATICAL TABLES. 



Number, 
or 


Circum- 


Circular 


Square. 


Cube. 


Square 


Cube 


Diameter. 


ference. 


Area. 




Root. 


Root. 


571 


1793-85 


256072.60 


326,041 


186,169,411 


23.896 


8.296 


572 


1796.99 


256970.31 


327,184 


187,149,248 


23.916 


8.301 


573 


1800.13 


257869.59 


328,329 


188,132,517 


23-937 


8.306 


574 


1803.27 


258770.45 


329,476 


189,119,224 


23.958 


8.3 II 


575 


1806.42 


259672.87 


330,625 


190,109,375 


23979 


8.315 


576 


1809.56 


260576.87 


331,776 


191,102,976 


24.000 


8.320 


577 


1812.80 


261482.43 


332,929 


192,100,033 


24.021 


8.325 


578 


1815.84 


262388.57 


334,084 


193,100,552 


24.042 


8.330 


579 


1818.98 


263298.28 


335,241 


194,104,539 


24.062 


8.335 


580 


1822.12 


264208.56 


336,400 


195,112,000 


24,083 


8.339 


581 


1825.26 


265120.46 


337,561 


196,122,941 


24.104 


8.344 


582 


1828.41 


266033.82 


338,724 


197,137,368 


24.125 


8.349 


583 


1831.55 


266948.82 


339,889 


198,155,287 


24.145 


8.354 


584 


1834.69 


267865.38 


341,056 


199,176,704 


24.166 


8.359 


585 


1837.83 


268783.57 


342,225 


200,201,625 


24,187 


8.363 


586 


1840.97 


269703.21 


343,396 


20i;23o,o56 


24.207 


8.368 


587 


1 844. 1 1 


270624.49 


344,569 


202,262,003 


24,228 


8.373 


588 


1847.26 


271547.33 


345,744 


203,297,472 


24.249 


8.378 


589 


1850.40 


272471.75 


346,921 


204,336,469 


24.269 


8.382 


590 


1853-54 


273397.74 


348,100 


205,379,000 


24.289 


8.387 


591 


1856.68 


274325.29 


349,281 


206,425,071 


1 24.310 


8.392 


592 


1859.82 


275254,42 


350,464 


207,474,688 


24.331 


8.397 


593 


1862.96 


276185.12 


351,649 


208,527,857 


24.351 


8.401 


594 


1 866. 11 


277117.39 


352,836 


209,584,584 


24.372 


8.406 


595 


1869.25 


278051.23 


354,025 


210,644,875 


24-393 


8.4II 


596 


1872.39 


278986.64 


355,216 


211,708,736 


24.413 


8.415 


597 


1875.53 


279923.62 


356,409 


212,776,173 


24.433 


8.420 


598 


1878.67 


280862.18 


357,604 


213,847,192 


24.454 


8.425 


599 


1881.81 


281802.30 


358,801 


214,921,799 


24.474 


8.429 


600 


1884.96 


282744.00 


360,000 


216,000,000 


24.495 


8.434 


601 


1888.10 


283687.26 


361,201 


217,081,801 


24.515 


8.439 


602 


1891.24 


284632.10 


362,404 


218,167,208 


24.536 


8.444 


603 


1894.38 


285578.50 


363,609 


219,256,227 


24.556 


8.448 


604 


1897.52 


286526.48 


364,816 


220,348,864 


24.576 


8.453 


605 


1900.66 


287476.03 


366,025 


221,445,125 


24.597 


8.458 


606 


1903.80 


288426.15 


367,236 


222,545,016 


i 24.617 


8.462 


607 


1906.95 


289379.84 


368,449 


223,648,543 


24.637 


8.467 


608 


1910.09 


290334.10 


369,664 


224,755,712 


24.658 


8.472 


609 


1913.23 


291289.93 


370,881 


225,866,529 


24.678 


8.476 


610 


1916.37 


2^)2247.34 


372,100 


226,981,000 


24.698 


8.481 


611 


1919.51 


293206.31 


373,321 


228,099,131 


24.718 


8.485 


612 


1922.65 


294166.85 


374,544 


229,220,928 


24.739 


8.490 


613 


1925.80 


295128.97 


375,769 


230,346,397 


24.758 


8.495 


614 


1928.94 


296092.65 


376,996 


231,475,544 


24.779 


8.499 


615 


1932.08 


297057.91 


378,225 


232,608,375 


24.799 


8.504 


616 


1935-22 


298024.74 


379,456 


233,744.896 


' 24.819 


8.509 


617 


1938.36 


298993.14 


380,689 


234,885,113 


24839 


8.513 


618 


1941.50 


299963.00 


381,924 


236,029,032 


1 24.859 


8.518 



NUMBERS, OR DIAMETERS OF CIRCLES, Sec 



79 



Nomber, 
Dbm«ter. 


CSrcuin- 
ferenc«. 


Circular 
Area. 


Square. 


Cube. 


Square 
Root. 


Cube 
Root. 


6io 

1 ^ 


1944.65 


300934,64 


383,161 


237,176,659 


24.879 


8.522 


620 


1947.79 


301907.76 


384,400 


238,628,000 


24.899 


8.527 


621 

1 


1950.93 


302S82.44 


3^5Mx 


239,483,061 


24.919 


8-532 


622 


1954-07 


303858.69 


386,884 


240,641,848 


24.939 


8.536 


6^3 


1957.21 


304836.51 


388,129 


241,804,367 


24.959 


8.541 


624 


1960.35 


305815.91 


389,376 


242,970,624 


24.980 


8.545 


625 


1963.50 


306796.87 


390,625 


244,140,625 


25.000 


8.549 


626 


1966.64 


307779.41 


391,876 


245,314,376 


25.019 


8.554 


627 


1969.78 


308763.41 


393,129 


246,491,883 


25.040 


8.559 


628 


1972.92 


309749.19 


394,384 


247,673,152 


25-059 


8.563 


629 


1976.06 


310736.44 


395,641 


248,858,189 


25.079 


8.568 


630 


1979.20 


311725.26 


396,900 


250,047,000 


25.099 


8.573 


631 


1982.34 


312715.64 


398,161 


25^239,591 


25.119 


8.577 


' 632 


1985.49 


313707.58 


399.424 


252,435,968 


25.139 


8.582 


^3S 


1988.63 


31470I.I4 


400,689 


253,636,137 


25.159 


8.586 


634 


1991.77 


315696.64 


401,956 


254,840,104 


25-179 


8.591 


635 


1994,91 


316692.91 


403,225 


256,047,875 


25.199 


8.595 


636 


1998.05 


31769I.15 


404,496 


257,259,456 


25.219 


8-599 


637 


2001.19 


318690.97 


405,769 


258,474,853 


25.239 


8.604 


638 


2004.34 


319692.35 


407,044 


259,694,072 


25.259 


8.609 


639 


2007.48 


320695.31 


408,321 


260,917,119 


25.278 


8.613 


640 


2010.62 


321699.84 


409,600 


262,144,000 


25.298 


8.6i8 


641 


2013.76 


322705.93 


410,881 


263,374,721 


25.318 


8.622 


642 


2016.90 


323713.60 


412,164 


264,609,288 


25.338 


8.627 


643 


2020.04 


324722.84 


413,449 


265,847,707 


25-357 


8.631 


644 


2023.19 


325733-65 


414,736 


267,089,984 


25-377 


8.636 


645 


2026.33 


326746.03 


416,025 


268,836,125 


25-397 


8.640 


646 


2029.47 


327759.98 


417,316 


269,586,136 


25.416 


8.644 


647 


2032.61 


328775.50 


418,609 


270,840,023 


25.436 


8.649 


648 


2035.76 


329792.60 


419,904 


272,097,792 


25.456 


8.653 


649 


2038.89 


330811.26 


421,201 


273,359,449 


25.475 


8.658 


650 


2042.04 


331831.50 


422,500 


274,625,000 


25-495 


8.662 


1 651 


2045.18 


332853.40 


423,801 


275,894,451 


25-515 


8.667 


652 


2048.32 


333876.68 


425,104 


277,167,808 


25-534 


8.671 


653 


2051.46 


334901.62 


426,409 


278,445,077 


25554 


8.676 


654 


2054.60 


335928.14 


427,716 


279,726,264 


25-573 


8.680 


' 655 


2057.74 


336956.23 


429,025 


281,011,375 


25.593 


8.684 


656 


2060.88 


3379S5.89 


430,336 


282,800,416 


25.612 


8.689 


657 


2064.03 


339017.12 


431,649 


283,593,393 


25.632 


8.693 


658 


2067.17 


340049.92 


432,964 


284,890,312 


25-651 


8.698 


659 


2070.31 


341084.29 


434,281 


286,191,179 


25.671 


8.702 


660 


2073.45 


342120.24 


435,600 


287,496,000 


25-690 


8.706 


661 . 


2076.59 


343157-75 


436,921 


288,804,781 


1 25.710 


8.711 


662 


2079.73 


344196.33 


438,244 


290,117,528 


25.720 


8.715 


663 


2082.88 


345237.49 


439,569 


291,434,247 


1 25.749 


8.719 


664 


2086.02 


346279.71 


440,896 


292,754,944 


25.768 


8.724 


665 


2089.16 


347323-51 


442,225 


294,079,625 


25-787 


8.728 


666 


2092.30 


^348368.88 


443,556 


295,408,296 


25.807 


8.733 



8o 



MATHEMATICAL TABLES. 



Number, 
or 


Circum- 


Circular 


Sauare. 


Cube. 


Square 


Cube 


Diameter. 


ference. 


Area. 


M^^i^KA^hM ^r» 




Root. 


Root. 


667 


2095.44 


349416.40 


444,889 


296,740,963 


25.826 


8.737 


668 


2098.58 


350464.32 


446,224 


298,077,632 


25.846 


8.742 


669 


2101.73 


351514-30 


447,561 


299,418,309 


25.865 


8.746 


670 


2104.87 


352566.06 


448,900 


300,763,000 


25.884 


8.750 


671 


2108.01 


353619.28 


450,241 


302,111,711 


25.904 


8.753 


672 


2III.15 


354674.07 


451,584 


303,464,448 


25-923 


8.759 


673 


2114.29 


355730-43 


452,929 


304,821,217 


25.942 


8.763 


674 


2117-43 


356788-37 


454,276 


306,182,024 


25.961 


8.768 


675 


2i2a58 


357847.87 


455,625 


307,546,875 


25.981 


8.772 


676 


2123.72 


358908.95 


456,976 


308,915,776 


26.000 


8.776 


677 


2126.86 


359971.59 


458,329 


310,288,733 


26.019 


8.781 


678 


2130.00 


361035.81 


459,684 


311,665,752 


26.038 


8.785 


679 


2133-14 


362101.60 


461,041 


313,046,839 


26.058 


8.789 


680 


2136.28 


363168.96 


462,400 


314,432,000 


26.077 


8.794 


681 


2139.42 


364237.88 


463,761 


315,821,241 


26.096 


8.798 


682 


2142.57 


365308.38 


465,124 


317,214,568 


26.115 


8.802 


683 


2145.71 


366380.40 


466,489 


318,611,987 


26.134 


8.807 


684 


2148.85 


367454.10 


467,856 


320,013,504 


26.153 


8.81 1 


685 


2151.99 


368529.31 


469,225 


321,419,125 


26.172 


8.815 


686 


2155-13 


369600.60 


470,596 


322,828,856 


26.192 


8.819 


687 


2158.27 


370684.45 


471,969 


324,242,703 


26.211 


8.824 


688 


2161.42 


371764.37 


473*344 


325,660,672 


26.229 


8.828 


689 


2164.56 


372845.87 


474,721 


327,082,769 


26.249 


8.832 


690 


2167.70 


373928.94 


476,100 


328,509,000 


26.268 


8.836 


1 691 


2170.84 


375013-57 


477,481 


329,939*371 


26.287 


8.841 


692 


2173.98 


376099.78 


478,864 


331,373,888 


26.306 


8.845 


693 


2177.12 


377187-56 


480,249 


332,812,557 


26.325 


8.849 


694 


2180.27 


378276.91 


481,636 


334,255,384 


26.344 


8.853 


695 


2183.41 


379367-83 


483,025 


335,702,375 


26.363 


8.858 


696 


2186.55 


380460.32 


484,416 


337,153,536 


26.382 


8.862 


697 


2189.69 


381554-38 


485,809 


338,608,873 


26.401 


8.866 


698 


2192.83 


382650.02 


487,204 


340,068,392 


26.419 


8.870 


699 


2195.97 


383747.22 


488,601 


341,532,099 


26.439 


8.875 


700 


2199.12 


384846.00 


490,000 


343,000,000 


26.457 


8.879 


701 


2202.26 


385949-52 


491,401 


344,472,101 


26.476 


8.883 


702 


2205.40 


387048.26 


492,804 


345,948,088 


26.495 


8.887 


703 


2208.54 


388151.74 


494,209 


347,428,927 


26.514 


8.892 


704 


2211.68 


389256.80 


495,616 


348,913,664 


26.533 


8.896 


705 


2214.82 


390363-43 


497,025 


350,402,625 


26.552 


8.900 


yo6 


2217.96 


391471-63 


498,436 


351,895,816 


26.571 


8.904 


707 


2221. II 


392581.40 


499,849 


353,393,243 


26.589 


8.908 


708 


2224.25 


393692.74 


501,264 


354,894,912 


26.608 


8.913 


709 


2227.39 


394805.65 


502,681 


356,400,829 


26.627 


8.917 


710 


2230.53 


395920.14 


504,100 


357,911,000 


26.644 


8.921 


711 


2233,67 


397036.19 


505,521 


359,425,431 


26.664 


8.925 


712 


2236.81 


398151.81 


506,944 


360,944,128 


26.683 


8.929 


713 


2239.96 


399273.01 


508,369 


362,467,097 


26.702 


8.934 


714 


2243.10 


400393.73 


509,796 


363,994,344 


26.721 


8.938 



NUMBERS, OR DIAMETERS OF CIRCLES, &c. 



8l 



I Kiunber, 

or 
Diameter. 



715 
716 

717 
718 

719 
720 

721 

722 

723 
724 

725 
726 

727 

728 

729 

731 
732 

733 
734 

735 
736 

737 

738 

739 
740 

741 
742 

743 

744 

745 
746 

747 
748 

749 
750 

751 
752 

753 

754 
755 
756 

757 
758 

759 
760 

761 

762 



Grcum- 
ference. 



2246.24 
2249.38 
2252.52 
2255.66 
2258.81 
2261.95 
2265.09 
226S.23 
2271.37 

2274-51 
2277.66 

2280.80 

2283.94 

2287.08 

2290.22 

2293.36 

2296.50 

2299.65 

2302.79 

2305.93 

2309-07 
2312.21 

2315.35 
2318.50 

2321.64 

2324.78 

2327.92 

2331.06 

2334.20 

2337.35 
2340.49 

2343.63 

2346.77 
2349.91 

2353.05 
2356.20 

2359.34 
2362.48 

2365.62 

2368.76 

2371.90 

2375.04 
2378.19 

2381.33 
2384.47 

2387.61 

2390.75 

2393-89 



Grcular 
Area. 



4OI516.II 
402640,02 

403765.50 
404892.54 

406021.16 
407151.36 
408283.32 
409416.45 
410551.25 
411687.93 
412825.87 
413965.24 
415106.06 
416249.43 

417393-76 
418539.66 
419687.12 
420836.14 
421986.78 
423138.96 
424292.71 
425442.03 
426604.93 

427763-39 
428923.43 

430085.04 

431248.21 

432412.96 

433579-28 

434747.17 
435916.63 

437087.66 

438260.26 

439434-48 
440610.18 
441787.50 
442966.38 
444146.84 
445328.86 
446512.46 

447697-63 
448884.37 

450072.68 

451262.56 

452454.01 

453647.04 
454841.63 

456037.87 



Square. 



5ii>225 
512,656 

514,089 
515*524 

516,961 
518,400 
519,841 
521,284 
522,729 

524>i76 
525*625 
527,076 
528,529 

529*984 
531,441 
532,900 

534,361 
535,824 
537,289 

538,756 
540,225 

541,696 

543,169 

544,644 
546,121 

547,600 

549,081 

550,564 
552,049 
553,536 

555,025 
556,516 
558,009 

559,504 
561,001 

562,500 

564,001 

565,504 
567,009 

568,516 

570,025 

571,536 

573,049 

574,564 
576,081 

577,600 

579,^21 

580,644 



Cube. 



365,525,875 
367,061,696 

368,601,813 

370,146,232 

371,694,959 
373,248,000 

374,805,361 

376,367,048 

377,933,067 

379,503,424 
381,078,125 

382,657,176 
384,240,583 
385,828,352 
387,420,489 
389,017,000 
390,617,891 
392,223,168 
393,832,837 
395,446,904 

397,065,375 
398,688,256 

400,315,553 

401,947,272 

403,583,419 
405,224,000 

406,869,021 

408,518,488 

410,172,407 

411,830,784 

413,493,625 

415,160,936 

416,832,723 

418,508,992 

420,189,749 

421,875,000 

423,564,751 
424,525,900 

426,957,777 

428,661,064 

430,368,875 

432,081,216 

433,798,093 
435,519,512 

437,245,479 
438,976,000 

440,711,081 

442,450,728 



Square 


Cube 


Root. 


Root. 


26.739 


8.942 


26.758 


8.946 


26.777 


8.950 


26.795 


8.954 


26.814 


8.959 


26.833 


8.963 


26.851 


8.967 


26.870 


8.971 


26.889 


8.975 


26.907 


8.979 


26.926 


^.983 


26.944 


8.988 


26.963 


8.992 


26.991 


8.996 


27.000 


9.000 


27.018 


9.004 


27.037 


9.008 


27.055 


9.012 


27.074 


9.016 


27.092 


9.020 


27. Ill 


9-023 


27.129 


9.029 


27.148 


9-033 


27.166 


9-037 


27.184 


9.041 


27.203 


9-045 


27.221 


9.049 


27.239 


9-053 


27.258 


9057 


27.276 


9.061 


27.295 


9.065 


27.313 


9.069 


27.331 


9-073 


27.349 


9-077 


27.368 


9.081 


27.386 


9.086 


27.404 


9.089 


27.423 


9.094 


27.441 


9.098 


27.459 


.9.102 


27.477 


9.106 


27.495 


9.109 


27-514 


9.II4 


27-532 


9. 1 18 


27.549 


9.122 


27.568 


9.126 


27.586 


9.129 


27.604 


9-134 



iV 



82 



MATHEMATICAL TABLES. 



Number, 

or 
Diameter. 


Circum- 
ference. 


Circular 
Area. 


Square. 


Cube. 


Square 
Root. 


Cube 
Root, 


763 


2397.04 


457235.53 


582,169 


444,194,947 


27.622 


9.138 


764 


2400.18 


458435.83 


583,696 


445,943,744 


27.640 


9.142 


765 


2403.32 


459635.71 


585*225 


447,697,125 


27.659 


9.146 


766 


2406.46 


460838.16 


586,756 


449,455,096 


27.677 


9.149 


767 


2409.60 


462042.18 


588,289 


451,217,663 


27.695 


9.154 


768 


2412.74 


463247.76 


589,824 


452,984,832 


27.713 


9.158 


769 


2415.98 


464454.92 


59i»36i 


454,756,609 


27.731 


9.162 


770 


2419.03 


465663.66 


592,900 


456,533,000 


27.749 


9.166 


771 


2422.17 


466873.96 


594,441 


458,314,011 


27.767 


9.169 


772 


2425-31 


468085.83 


595,984 


460,099,648 


27.785 


9.173 


77^ 


2428.45 


469299.27 


597,529 


461,889,917 


27.803 


9.177 


774 


2431-59 


470514.29 


599,076 


463,684,824 


27.821 


9.181 


775 


2434-74 


471730.87 


600,625 


465,484,375 


27.839 


9.185 


776 


2437.88 


47^2949.03 


602,176 


467,288,576 


27.857 


9.189 


777 


2441.02 


474168.75 


603,729 


469,097,433 


27.875 


9.193 


778 


2444.16 


475396.05 


605,284 


470,910,952 


27.893 


9.197 


779 


2447.30 


476612.92 


606,841 


472,729,139 


27.910 


9.201 


780 


2450.44 


477837.36 


608,400 


474,552,000 


27.928 


9.205 


781 


2453.58 


479063.36 


609,961 


476,379,541 


27.946 


9.209 


782 


2456.73 


480290.94 


611,524 


478,211,768 


27.964 


9.213 


783 


2459.87 


481520.10 


613,089 


480,048,687 


27.982 


9.217 


784 


2463.01 


482750.82 


614,656 


481,890,304 


28.000 


9.221 


785 


2466.15 


483983.11 


616,225 


483,736,025 


28.017 


9.225 


786 


2469.29 


485216.97 


617,796 


485,587,656 


28.036 


9.229 


787 


2472.43 


486452.41 


619,369 


487,443,403 


28.053 


9.233 


788 


2475.48 


487689.73 


620,944 


489,303,872 


28.071 


9.237 


789 


2478.72 


488927.99 


622,521 


491,169,069 


28.089 


9.240 


790 


2481.86 


490168.14 


624,100 


493,039,000 


28.107 


9.244 


791 


2485.00 


491409.85 


625,681 


494,913,671 


28.125 


9.248 


792 


2488.14 


492653.14 


627,264 


496,793,088 


28.142 


9.252 


793 


2491.28 


493898.20 


628,849 


498,677,257 


28.160 


9.256 


.794 


2494.43 


495144.43 


630,436 


500,566,184 


28.178 


9.260 


795 


2497.57 


496392.43 


632,025 


502,459,875 


28.196 


9.264 


796 


2500.71 


497648.40 


633,616 


504,358,336 


28.213 


9.268 


797 


2503.85 


498893.14 


635,209 


506,261,573 


28.231 


9.271 


798 


2506.99 


500145.86 


636,804 


508,169,592 


28.249 


9.275 


799 


2510.13 


501400.14 


638,401 


510,082,399 


28.266 


9.279 


800 


2513.28 


502656.00 


640,000 


512,000,000 


28.284 


9.283 


801 


2516.42 


503913.42 


641,601 


513,922,401 


28.302 


9.287 


802 


2519.56 


505172.43 


643,204 


515,849,608 


28.319 


9.291 


803 


2522.70 


506432.98 


644,809 


517,781,627 


28.337 


9.295 


804 


2525.84 


507655.52 


646,416 


519,718,464 


28.355 


9.299 


805 


2528.98 


508958.83 


648,025 


521,660,125 


28.372 


9.302 


806 


2532.12 


510224.II 


649,636 


523,606,616 


28.390 


9.306 


807 


2535.27 


511490.96 


651,249 


525,557,943 


28.408 


9.310 


808 


2538.41 


512759.38 


652,864 


527,514,112 


28.425 


9.314 


809 


2541.55 


514029.37 


654,481 


529,474,129 


28.443 


9.318 


810 


2544.09 


515300.94 


656,100 


531,441,000 


28.460 


9.321 



NUMBERS, OR DIAMETERS OF CIRCLES, &c. 



83 



Number, 

or 


Crcum- 


Circular 


Square. 


Cube. 


Square 


Cube 


Dianwtw. 


ference. 


Area. 






Root. 


Root. 


i 811 


2547.83 


516574.07 


657,721 


533,411,731 


28.478 


9-325 


812 


2550-97 


517848.77 


659,344 


535,387,328 


28.496 


9-329 


813 


2554-12 


519125.05 


660,969 


537,366,797 


28.513 


9-333 


814 


2557.26 


520402.85 


662,596 


539,353,144 


28.531 


9-337 


815 


2560.40 


521682.31 


664,225 


541,343,375 


28.548 


9-341 


S16 


2563.54 


522663.30 


665,856 


543,338,496 


28.566 


9-345 


817 


2566.68 


524245.86 


667,489 


545,338,513 


28.583 


9-348 


818 


2569.82 


525529.98 


669,124 


547,343,432 


28.601 


9.352 


819 


2572.97 


526815,68 


670,761 


549,353,259 


28.618 


9-356 


820 


2576.11 


528102.96 


672,400 


551,368,000 


28.636 


9.360 


821 


2579.25 


529391.80 


674,041 


553,387,6^1 


28.653 


9-364 


822 


2582.39 


530682.21 


675,684 


555,412,248 


28.670 


9-367 


823 


2585.53 


531974.39 


677,329 


557,441,767 


28.688 


9.371 


824 


2588.64 


533267.75 


678,976 


559,476,224 


28.705 


9-375 


825 


2591.82 


534562.87 


680,625 


561,515,625 


28.723 


9.379 


826 

1 


2594.96 


535859.57 


682,276 


563,559,976 


28.740 


9-383 


8a7 


2598.10 


537159-83 


683,929 


565,609,283 


28.758 


9-386 


828 


2601.24 


538457-62 


685,584 


567,663,552 


28.775 


9.390 


829 


2604-38 


539759.08 


687,241 


569,722,789 


28.792 


9-394 


830 


2607.52 


541062.06 


688,900 


571,787,000 


28.810 


9.398 


831 


2610.66 


542366.60 


690,561 


573.856,191 


28.827 


9.401 


83* 


2613.81 


543672.72 


692,224 


575,930,368 


28.844 


9.405 


833 


2616.95 


544980.52 


693,889 


578,009,537 


28.862 


9.409 


834 


2620.09 


546289.68 


695,556 


580,093,704 


28.879 


9.413 


i 835 


2623.23 


547600.51 


697,225 


582,182,875 


28.896 


9.417 


836 


2626.37 


548912.91 


698,896 


584,277,056 


28.914 


9.420 


837 


2629.51 


550226.89 


700,569 


586,376,253 


28.931 


9.424 


838 


2632.64 


551542.43 


702,244 


588,480,472 


28.948 


9.428 


839 


263S.80 


552859-58 


703,921 


590,589,719 


28.965 


9-432 


840 


2638.94 


554178.24 


705,600 


592,704,000 


28.983 


9-435 


84, 


2642.08 


555498.49 


707,281 


594,823,321 


29.000 


9-439 


, 842 


2645.22 


556820.32 


708,964 


596,947,688 


29.017 


9-443 


! 843 


2648.36 


558143.72 


710,649 


599,077,107 


29.034 


9-447 


844 


2651.51 


559468.69 


712,336 


601,211,584 


29.052 


9-450 


84s 


2654.65 


560795.23 


714,025 


603,351,125 


29.069 


9-454 


846 


2657.79 


562123.34 


715,716 


605,495,736 


29.086 


9.458 


, 847 


2660.93 


563456.82 


717,409 


607,645,423 


29.103 


9.461 


1 848 


2664,07 


564784.28 


719,104 


609,800,192 


29.120 


9.465 


849 


2667.21 


566117.IO 


720,801 


611,960,049 


29.138 


9-469 


, 850 


2670.36 


567451-59 


722,500 


614,125,000 


29.155 


9-473 


1 851 


2673.50 


568787.46 


724,201 


616,295,051 


29.172 


9-476 


852 


' 2676.64 


570125.00 


725,904 


618,470,208 


29.189 


9.480 


' 853 


2679.78 


571464.10 


727,609 


620,650,477 


29.206 


9-483 


' 854 


2682.92 


572804.78 


729,316 


622,835,864 


29.223 


9.487 


85s 


2686.06 


574147.03 


731,025 


625,026,375 


29.240 


9.491 


' 856 


2689.20 


575490.85 


732,736 


627,222,016 


29-257 


9-495 


' ^57 


2692.35 


576836.24 


734,449 


629,422,793 


29-274 


9-499 


i 858 


2695.49 


578183.20 


736,164 


631,628,712 


29.292 


9.502 



84 



MATHEMATICAL TABLES. 



Number, 
or 


Circum- 


Circular 


Square. 


Cube. 


Square 


Cube 


Diameter. 


ference. 


Area. 






Root. 


Root. 


859 


2698.63 


579531.73 


737,881 


633,839,779 


29.309 


9.506 


860 


2701.77 


580881.84 


739,600 


636,056,000 


29.326 


9-509 


861 


2704.91 


582233.51 


741,321 


638,277,381 


29.343 


9.513 


862 


2708.05 


583586.75 


743,044 


640,503,928 


29.360 


9-517 


863 


2711.20 


584941.57 


744,769 


642,735,647 


29.377 


9.520 


864 


2714-34 


586297.95 


746,496 


644,972,544 


29394 


9-524 


865 


2717.48 


587655-91 


748,225 


647,214,625 


29.411 


9.528 


866 


2720.66 


589015.41 


749,956 


649,461,896 


29.428 


9532 


867 


2723.76 


590376.54 


751,689 


651,714,363 


29-445 


9-535 


868 


2726.90 


591739.20 


753,424 


653,972,032 


29.462 


9.539 


869 


2730.05 


5<>3 103-44 


755,161 


656,234,909 


29.479 


9-543 


870 


2733-19 


594469.26 


756,900 


658,503,000 


29.496 


9-546 


871 


2736.33 


595836.44 


758,641 


660,776,311 


29.513 


9-550 


872 


2739.87 


597205.59 


760,384 


663,054,848 


29.529 


9-554 


873 


2742.61 


598576.91 


762,129 


665,338,617 


29.546 


9-557 


874 


2745-75 


599948.21 


763,876 


667,627,624 


29.563 


9.561 


875 


2748.90 


601321.87 


765,625 


669,921,875 


29.580 


9-565 


876 


2752.04 


602697,11 


767,376 


672,221,376 


29-597 


9.568 


877 


2755-18 


604073.91 


769,129 


674,526,133 


29.614 


9-572 


878 


2758.32 


605451.49 


770,884 


676,836,152 


29.631 


9.575 


879 


2761.46 


606832.24 


772,641 


679,151,439 


29.648 


9.579 


880 


2764.60 


608213.76 


774,400 


681,472,000 


29.665 


9.583 


881 


2767.74 


609596.84 


776,161 


683,797,841 


29.682 


9.586 


882 


2770.89 


610981.50 


777,924 


686,128,968 


29.698 


9-590 


883 


2774.03 


612367.74 


779,689 


688,465,387 


29-715 


9.594 


884 


2777.17 


613755-54 


781,456 


690,807,104 


29.732 


9-597 


885 


2780.31 


615144.91 


783,225 


693,154,125 


29-749 


9.601 


886 


2783-45 


616535-85 


784,996 


695,506,456 


29.766 


9.604 


887 


2786.59 


617928.37 


786,769 


697,864,103 


29.782 


9.608 


888 


2789-75 


619322.45 


788,544 


700,227,072 


29.799 


9.612 


889 


2792.88 


620718.11 


790,321 


702,595,369 


29.816 


9-615 


890 


2796.02 


622115,34 


792,100 


704,969,000 


29.833 


9.619 


891 


2799.16 


623514.13 


793,881 


707,347,971 


29.850 


9.623 


892 


2802.30 


624914.50 


795,664 


709,732,288 


29.866 


9.626 


893 


2805.44 


626316.44 


797,449 


712,121,957 


29.883 


9.630 


894 


2808.59 


627719-95 


799,236 


714,516,984 


29.900 


9.633 


895 


2811.73 


629120.35 


801,025 


716,917,375 


29.916 


9-637 


896 


2814.87 


630531.68 


802,816 


719,323,136 


29933 


9.640 


897 


2818.82 


631939-90 


804,609 


721,734,273 


29.950 


9.644 


898 


2821.15 


633349-70 


806,404 


724,150,792 


29.967 


9.648 


899 


2824.29 


634768.13 


808,201 


726,572,699 


29.983 


9-651 


900 


2827.44 


636174.00 


810,000 


729,000,000 


30.000 


9.655 


901 


2830.58 


637588.50 


811,804 


731,432,701 


30.017 


9.658 


902 


2833.72 


639004.58 


813,604 


733,870,808 


30.033 


9.662 


903 


2836.86 


640422.22 


815,409 


736,314,327 


30.050 


9.666 


904 


2840.00 


641841.44 


817,216 


738,763,264 


30.066 


9.669 


905 


2843.14 


643262,23 


819,025 


741,217,625 


30.083 


9.673 


906 


2846.28 


644684.74 


820,836 


743,677,416 


30.100 


9.676 



NUMBERS, OR DIAMETERS OF CIRCLES, &c. 



85 



Xumber, 
or 


Grcum- 




ference. 


907 


2849.43 


908 


2852.57 


909 


2855-71 


910 


2858.85 


911 


2861.99 


912 


2865.13 


913 


2868.29 


914 


2871.42 


915 


2874.56 


916 


2877.70 


917 


2880.84 


918 


2883.98 


919 


2887.13 


920 


2890.27 


921 


2893.41 


922 


2896.55 


923 


2899.69 


924 


2902.83 


925 


2905.98 


926 


2909.12 


927 


2912.26 


928 


2915.40 


929 


2918.54 


930 


2921.68 


931 


2924.82 


932 


2927.97 


933 


2931. II 


934 


2934.25 


935 


2937.39 


936 


2940.53 


937 


2943.67 


938 


2946.82 


939 


2949.96 


940 


2953.10 


941 


2956.24 


942 


2959.38 


943 


2962.43 


944 


2965.67 


945 


2968.81 


946 


2971.95 


947 


2975.09 


948 


2978.23 


949 


2981.37 


950 


2984.52 


951 


2987.66 


952 


2990.72 


953 


299394 


954 


2997.08 



Ctrcttlar 
Area. 


Square. 


646108.52 


822,649 


64753402 


824,464 


648961.09 


826,281 


650389.74 


828,100 


651819.95 


829,921 


653251-73 


831,744 


654689.09 


833,569 


656120.81 


835,396 


657556.51 


837,225 


658994.58 


839,056 


660432.22 


840,889 


661875.42 


842,724 


663318.20 


844,561 


664762.56 


846,400 


666208.48 


848,241 


667655-97 


850,084 


669IOI.61 


851,929 


670555-67 


853,776 


672007.87 


855,625 


673461.65 


857,476 


674916.99 


859,329 


676373-91 


861,184 


677832.40 


863,041 


679292.46 


864,900 


680754.08 


866,761 


682217.30 


868,624 


683682.06 


870,489 


685148.40 


872,356 


686616.31 


874,225 


688085.79 


876,096 


689556.85 


877,969 


691029.47 


879,844 


692503.67 


881,721 


693979-44 


883,600 


695456.77 


885,481 


696935-68 


887,364 


698416.14 


889,249 


699898.21 


891,136 


701381.83 


893,025 


702867.02 


894,916 


704350.25 


896,809 


705841.80 


898,704 


707332.02 


900,601 


708023.50 


902,500 


710316.54 


904,401 


7I181I.16 


906,304 


713307-34 


908,209 


714805.10 


910,116 



Cube. 



746,142,643 
748,613,312 
751,089,429 
753,571,000 
756,058,031 
758,550,528 
761,048,497 

763,551,944 
766,060,875 

768,575,296 

771,095,213 

773,620,632 

776,151,559 
778,688,000 

781,229,961 

783,777,448 
786,330,467 
788,889,024 
791,453,125 
794,022,776 

796,597,983 
799,178,752 
801,765,089 

804,357,000 
806,954,491 

809,557,568 
812,166,237 
814,780,504 
817,400,375 
820,025,856 
822,656,953 
825,293,672 
827,936,019 
830,584,000 
833,237,621 
835,896,888 
838,561,807 
841,232,384 
843,908,625 

846,590,536 
849,278,123 

851,971,392 

854,670,349 
857,375,000 

860,085,351 

862,801,408 

865,523,177 
868,250,664 



Square 


Cube 


Root. 


Root. 


30.116 


9.680 


30.133 


9.683 


30.150 


9.687 


30.163 


9.690 


30.183 


9-694 


30.199 


9.698 


30.216 


9.701 


30.232 


9-705 


30.249 


9.708 


30.265 


9.712 


30.282 


9.715 


30.298 


9.718 


30.315 


9.722 


30.331 


9.726 


30.348 


9.729 


30.364 


9.733 


30.381 


9-736 


30.397 


9.740 


30.414 


9-743 


30.430 


9-747 . 


30.447 


9-750 


30.463 


9-754 


30.479 


9-757 


30.496 


9.761 


30.512 


9.764 


30.529 


9.768 


30.545 


9.771 


30.561 


9.775 


30.578 


9.778 


30.594 


9.783 


30.610 


9.785 


30.627 


9.789 


30.643 


9.792 


30.659 


9.796 


30.676 


9.799 


30.692 


9.803 


30.708 


9.806 


30.724 


9.810 


30.741 


9.813 


30.757 


9.817 


30.773 


9.820 


30.790 


9.823 


30.806 


9.827 


30.822 


9.830 


30.838 


9.834 


30.854 


9.837 


30.871 


9.841 


30.887 


9.844 



86 



MATHEMATICAL TABLES. 



j Number, 
or 


Circum- 


Circular 


Sauaie. 


Cube. 


Square 


Cube 




Diameter. 


ference. 


Area. 


»i^*i **•*• *p* 




Root. 


Root. 




955 


3000.22 


716304.43 


912,025 


870,983,875 


30.903 


9.848 


956 


3003-36 


717805.33 


913,936 


873,722,816 


30.919 


9.851 




957 


3006.51 


719307.80 


915,849 


876,467,493 


30.935 


9.854 




958 


3009.65 


720811.84 


917,764 


879,217,912 


30.951 


9.858 




959 


3012.79 


722317.45 


919,681 


881,974,079 


30.968 


9.861 




960 


3015-93 


723824.64 


921,600 


884,736,000 


30.984 


9.865 




961 


3019.07 


725333-39 


923»52i 


887,503,681 


31.000 


9.868 




962 


3022.21 


726843.71 


925,444 


890,277,128 


31.016 


9.872 




963 


3025.36 


728355-61 


927,369 


893,056,347 


31.032 


9-875 




964 


3028.50 


729869.07 


929,296 


895,841,344 


31.048 


9.878 




965 


3031.64 


73 1384. 1 1 


931,225 


898,632,125 


31.064 


9.881 




966 


3034.78 


732900.72 


933,156 


901,428,696 


31.080 


9.885 




967 


3037.92 


734418.90 


935,089 


904,231,063 


31.097 


9.889 




968 


3041.06 


735938.64 


937,024 


907,039,232 


3^'^^3 


9.892 




969 


3044.21 


737459.96 


938,961 


909,853,209 


31.129 


9.895 




970 


3047.35 


738982.86 


940,900 


912,673,000 


31.145 


9.899 




971 


3050.49 


740507.32 


942,841 


915,498,611 


31.161 


9.902 




972 


3053.63 


742033.35 


944,784 


918,330,048 


31.177 


9.906 




973 


3056.77 


743560.95 


946,729 


921,167,317 


31.193 


9-909 




• 974 


3059.91 


745090.13 


948,676 


924,010,424 


31.209 


9.912 




975 


3063.06 


746620.87 


950,625 


926,859,375 


31.225 


9.916 




976 


3066.20 


748153.19 


952,576 


929,714,176 


31.241 


9.919 




977 


3069.36 


749687.07 


954,529 


932,574,833 


31-257 


9923 




978 


3072.48 


751222.53 


956,484 


935,441,352 


31.273 


9.926 




979 


3075.62 


752759-56 


958,441 


938,313,739 


31.289 


9.929 




980 


3078.76 


754298.16 


960,400 


941,192,000 


31-305 


9-933 




981 


3081.90 


755838.32 


962,361 


944,076,141 


31-321 


9-936 




982 


3085.05 


757380.06 


964,324 


946,966,168 


31.337 


9.940 




983 


3088.19 


758923.38 


966,289 


949,862,087 


31.353 


9-943 




984 


3091.33 


760468.26 


968,256 


952,763,904 


31.369 


9.946 




985 


3094.47 


762014.71 


970,225 


955,671,625 


31-385 


9.950 




986 


3097-61 


763562.73 


972,196 


958,585,256 


31.401 


9-953 




987 


3100.75 


765119-33 


974,169 


961,504,803 


31.416 


9-956 




988 


3103.96 


766663.49 


976,144 


964,430,272 


31.432 


9.960 




989 


3107.04 


768216.23 


978,121 


967,361,669 


31.448 


9963 




990 


3IIO.18 


769770.54 


980,100 


970,299,000 


31.464 


9.966 




991 


3113.32 


771326.41 


982,081 


973,242,271 


31.480 


9.970 




992 


3116.46 


772883.86 


984,064 


976,191,488 


31.496 


9.973 




993 


3119.60 


774442.88 


986,049 


979,146,657 


31.512 


9.977 




994 


3122.75 


776003.47 


988,036 


982,107,784 


31.528 


9.980 




995 


3125.89 


777565.63 


990,025 


985,074,875 


31.544 


9.983 




996 


3129.03 


779129.36 


992,016 


988,047,936 


31.559 


9.987 




997 


3132.17 


780694.66 


994,009 


991,026,973 


31.575 


9.990 




998 


3135-11 


782261.54 


996,004 


994,011,992 


31-591 


9-993 




999 


3138.45 


783829.98 


998,001 


997,002,999 


31.607 


9.997 




1000 


3141.60 


785400.00 


1,000,000 


1,000,000,000 


31.623 


10.000 





CIRCLES: — DIAMETER, CIRCUMFERENCE, &C. 



87 



TABLE No. IV. CIRCLES:— DIAMETER, CIRCUMFERENCE, 
AREA, AND SIDE OF EQUAL SQUARE. 



Diamecer. 


Grcum- 
fereoce. 


Area. 


Side of 

Equal Souare 

(Square Root 

of Area). 


Diameter. 


Circum- 
ference. 


Area. 


Side of 

Equal Square 

(Square Root 

of Area). 


S/«6 

H 

1 t 
9/16 

) H 
'J" 


.1963 
.3927 

-5890 

.7854 

.9817 

I.I781 

. 1.3744 
1.5708 

1.7771 

1.9635 
2.1598 

2.3562 

2.5525 
2.7489 

2.9452 


.00307 

.01227 

.02761 

.04909 

.07670 

.1104 

.1503 

.1963 

.2485 
.3068 

.3712 

.4417 
.5185 

.6013 

.6903 


.0553 
.1107 

.1661 

.2215 

.2770 

.3323 
.3877 

.4431 
.4984 

.5539 
.6092 

.6646 

.7200 

.7754 
.8308 


3 , 

3'A 

3 3/.« 

3X 

'J' 
37/16 

3 9A6 

3H 

It 

3'5/i6 


9.4248 
9.62 II 

9.8175 
10.014 

10.210 

10.406 

10.602 

10.799 

10.995 
II. 191 

11.388 

11.584 

II.781 

11.977 

12.173 
12.369 


7.0686 
7.3662 
7.6699 

7.9798 
8.2957 
8.6180 
8.9462 
9.2807 
9.6211 
9.9680 

ia320 
10.679 
11.044 
II.4I6 

11.793 

12,177 


2.6586 
2.7140 

2.7694 
2.8248 
2.8801 

2.9355 
2.9909 

3.0463 
3.1017 

3.1571 
3.2124 

3-2678 

3-3232 
3.3786 

3.4340 

3.4894 


I 

I 5/:6 

i 9/16 

I"Vr6 


3.1416 

3.3379 

3.5343 
3-7306 
3.9270 

4.1233 
4.3197 
4.5160 

4-7124 

4-9087 
5. 105 1 

5.3014 
5-4978 
S.694I 
t.8905 
60868 


.7854 

.8866 
.9940 

1.1075 
1.227 1 

1.3530 
1.4848 
1.6229 
1.767 1 

1.9175 
2.0739 

2.2365 

2.4052 

2.5800 

2.761 1 

2.9483 


.8862 
.9416 

.9969 
1.0524 

I.IOI7 

I.163I 

I.2185 

1.2739 

1.3293 

1.3847 
1. 4401 

1.4955 
1.5508 

1.6062 

I.6616 

I.7170 


43/16 

4X 

4 7/,6 

49/16 

4'S/i6 


12.566 
12.762 

12.959 
13-155 
13-351 
13-547 
13-744 
13.940 

14.137 
14.333 
14.529 
14.725 
14.922 
15.119 

15.315 
15.511 


12.566 
12.962 

13-364 
13-772 
14186 
14606 

15.033 
15-465 
15.904 

16.349 

16.800 

17.257 

17.720 
18.190 
18.665 
19.147 


3.5448 
3.6002 

3.6555 
3-7109 

3.7663 

3.8217 

3.8771 

3-9325 
3.9880 

4.0434 
4.0987 
4.1541 
4.2095 

42648 
43202 
4-3756 


2 
2 V16 

2 3/,6 

 2 5/,6 

! 2^ 
2 7/.6 
2>^ 
2 9/x6 

2;i 
2»»yi6 

2'3/,6 
2« 
2'5/«6 1 


6.2832 

6^795 
6.6759 

6.J722 

7.0686 

7-2*49 

7-4*13 
7.6576 

7.8540 
8.0533 
8.2^67 
8.4430 
8.6314 
8.83f7 
9.0311 


3.1416 

3.3380 

3.5465 

3.7584 
3.9760 

4.2000 
4.4302 
3.7066 
4.9087 

5-1573 

5-4119 

5.6723 

5-9395 
6.2126 

6.4918 

6.7772 


1.7724 
1.8278 
1.8831 

1.9385 

1.9939 
2.0493 

2.1047 

2. 1 601 

2.2155 

2.2709 

2.3262 

2.3816 

2.4370 

2.4924 
2.5478 
2.6032 


5 

5 '/16 

5 3/,6 

59/16 

St 

5'5/i6 


15.708 
15.904 
16.100 
16.296 

16493 
16.689 

16.886 

17.082 

17.278 

17.474 
17-671 
17.867 
18.064 
18.261 

18.457 
18.653 


19.635 

20.129 
20.629 

21.135 

21.647 
22.166 
22.690 
23.221 

23.758 

24301 
24850 

25.406 

25.967 
26.535 

27.108 

27.688 


4.4310 
44864 

4-5417 

4-5971 
46525 

4.7079 
4.7633 
48187 

4.8741 

4-9295 
49848 

5.0402 

5-0956 
5.1510 
5.2064 
5.2618 



MATHEMATICAL TABLES. 



CIRCLES: — DIAMETER, CIRCUMFERENCE, &C. 



89 









Side of 








Side of 


.Diameter. 


Circum- 
ference. 


Area. 


Equal Sauare 

(Square Root 

of Area). 


Diameter. 


Circum- 
ference. 


Area. 


Equal Sauare 

(Square Root 

of Area}. 


iS 


56.548 


254.469 


15-951 


24 


75-398 


452.390 


21.268 


liyi 


56.941 


258.016 


16.062 


24}i 


75-791 


457-115 


21.379 


i&% 


57.334 


261.587 


16.173 


24X 


76.183 


461.864 


21 490 


tSH 


57.726 


265.182 


16.283 


24^ 


76.576 


466.638 


21.601 


18^ 


58.119 


268.803 


16.394 


24K 


76.969 


471.436 


21.712 


18H 


58.512 


272.447 


16.505 


24H 


77.361 


476.259 


21.822 


ii^ 


58.905 


276.117 


16.616 


24«' 


77-7SA- 


481.106 


21.933 


i»H 


59.297 


279.811 


16.727 


24;^ 


78.147 


485.978 


22.044 


19 


59.690 


283.529 


16.837 


25 


78.540 


490.875 


22.155 


19X 


60.083 


287.272 


16.948 


25 >i 


78.932 


495-796 


22.265 


19V 


60.475 


291.039 


17.060 


25 X 


79-325 


500.741 


22.376 


t9H 


6a868 


294.831 


17.170 


25^ 


79718 


505.711 


22.487 


•9>i 


61.261 


298.648 


17.280 


25 K 


80.110 


510.706 


22.598 


19^ 


61.653 


302.489 


17.391 


2SH 


80.503 


515-725 


22.709 


19^ 


62.046 


306.355 


17.502 


2sH 


80.896 


520.769 


22.819 


i9;< 


62.439 


310.245 


17.613 


25^ 


81.288 


525.837 


22.930 


1 - 
20 


62.832 


314.160 


17.724 


26 


81.681 


530.930 


23.041 


20X 


63.224 


318.099 


17.834 


26% 


82.074 


536.047 


23.152 


20X 


63.617 


322.063 


17.945 


26X 


82.467 


541.189 


23.062 


20^ 


64.010 


326.05 I 


18.056 


26^ 


82.859 


546.356 


23.373 


70}4 


64.402 


330.064 


18.167 


26X 


83.252 


551-547 


23484 


20>i 


64-795 


334-101 


18.277 


26^ 


83.645 


556.762 


23.595 


204i^ 


65.188 


338.163 


18.388 


26X 


84.037 


562.002 


23.708 


20;< 


65.580 


342.250 


18.499 


26^ 


84.430 


567.267 


23.816 


21 


65.973 


346.361 


18.610 


27 


84-823 


572.556 


23.927 


21^ 


66.366 


350.497 


18.721 


27% 


85.215 


577.870 


24.038 


21X 


66.759 


354.657 


18.831 


27H 


85.608 


583.208 


24.149 


21H 


67.151 


358.841 


18.942 


27H 


80.001 


588.571 


24.259 


21X 


67.544 


363.05 1 


19053 


27 K 


86.394 


593.958 


24.370 


21^ 


67.937 


367.284 


19.164 


27H 


86.786 


599370 


24481 


. 21^ 


68.329 


371.543 


19.274 


27H 


87.179 


604.807 


24.592 


. 21^ 


68.722 


375.826 


19.385 


27% 


87.572 


610.268 


24.703 


1 
22 


69.115 


380.133 


19.496 


28 


87.964 


615753 


24.813 


22X 


69.507 


384-465 


19.607 


28^ 


88.357 


621.263 


24924 


22 V 


69.900 


388.822 


19.718 


28X 


88.750 


626.798 


25.035 


22^ 


70.293 


393.203 


19.828 


28^ 


89.142 


632.357 


25.146 


5 22>i 


70.686 


397.608 


19.939 


28X 


89.535 


637.941 


25.256 


22K 


ji.orj^ 


402.038 


20.050 


28;^ 


89.928 


643.594 


25.367 


-i^H 


7^-471 


406.493 


20.161 


28^ 


90.321 


649.182 


25.478 


22 J< 


71.864 


410.972 


20.271 


28?^ 


90.713 


654-839 


25.589 


23 


72.256 


415.476 


20.382 


29 


91.106 


660.521 


25.699 


23^ 


72.649 


420.004 


20.493 


29^ 


91.499 


666.227 


25.810 


23X 


73.042 


424-557 


20.604 


29X 


91.891 


671.958 


25.921 


23>i 


73-434 


429.135 


20.715 


29^ 


92.284 


677.714 


26.032 , 


23;^ 


73.827 


433.731 


20.825 


2rA 


92.677 


683.494 


26.143 


23H 


74.220 


438.363 


20.936 


29H 


93.069 


689298 


26.253 . 


23JK' 


74.613 


443.014 


21.047 


293^ 


93-462 


695.128 


26.364 


23^ 


75.005 


447.699 


21.158 


29;^ 


93855 


700.981 


26.478 



rX 



MATHEMATICAL TABLES. 









Sid= or 








Side of 


Dinmrtcr, 


^— 


a™. 


Equal Smian 

iSquirrlt™. 




. 


a™. 


iSquilTRool 

Of Ana). 


30 


94.248 


706.860 


26.586 


^^. 


113.097 


10.7.87 


31-903 


30 Ji 


94.640 


712.762 


26.696 


36 J^ 


113.490 


1024.9s 


32.014 


30^ 


95-033 


718.690 


26.807 


36X 


113-883 


1032.06 


32.124 


30« 


95.426 


724.641 


26.918 


36« 


114.27; 


1039-19 


32-235 


3o;i 


95-818 


730.618 


27.029 


36X 


1 14.668 


1046.3 s 


32-349 


3o« 


96.211 


736.619 


27.139 


i(>H 


115.061 


■053-52 
1060.73 


32.457 


742.644 


27.250 


3^X 


115.453 


32-567 




748.694 


27.361 


36^ 


MS.846 


1067.95 


32.678 


7'; 4.760 


27.472 


37 


1 16.239 


1075.21 


32.789 


760.868 


27.583 


37 fi 


1 [6,631 


1082.48 


32,900 


766.992 


27.693 


37X 


117,024 


1089.79 


33-011 


773.140 


27,804 


37H 


117.417 


1097.11 


33.021 


779'3i3 


27.915 


y}% 


117,810 


1104.46 


33.232 


78;.sio 


28.026 


37H 




1U..84 


33-343 


791.732 


28. 136 


yjH 


[•.l$l 


1 1 [9.24 


33.454 




797.978 


28.247 


37Pi 


1126.66 


33564 


804.249 


28.358 


3^ . 


119,380 


1.34.11 


33.675 


810.545 


28.469 


38 H 


119-773 


1.41.59 


33786 


816.865 


28.580 


38y 


120.166 




33.897 


823.209 


28.69. 


3m 


120,558 




34.008 


829.578 


28.801 


38 Ji 


120,951 




34-118 


835-972 


28.9.2 


iSH 


121.344 


1171-73 


34-229 


842.390 


29.023 


3SH 


121.737 


1179.32 


34.340 




848.833 


29.133 


3i^ 


122.1Z9 


1.86.94 


34-45" 


855.30 


29.244 


39 


122,522 


1194-59 


34.561 


861.79 


^9.355 


39>i 


122,9.5 


1202.26 


34.672 


868.30 


29,466 


39X 


123.307 


1209.95 


34.783 


874.84 


^9-577 


39^ 


123.700 


1217.67 


34.894 


881.41 


29,687 


39^ 


124.093 
124-485 


1225.42 


35-005 J 


888.00 


29,798 


39H 


1233-18 


35-115 


894.61 


29.909 


39>^ 


.24.878 


! 240.98 


35-226 ' 




901.25 


30.020 


39H 


125.271 


1248,79 


35-337 1 


007.92 


3o->3' 


40 


125.664 


.256.64 


35-448 


914.61 


30,241 


40H 


126.056 


1264.50 


3S-SSS 


921.32 


30.352 


40V 


126.449 


1272.39 


35.669 


928.06 


30.463 


^oH 


126.842 


.280.31 


35780 


934-82 


30574 


Ao'A 


127.334 


1288.25 


35-891 


941.60 


30.684 


AoH 


127.627 


1296.21 


36.002 


948.41 


30.79s 


40K 


128.020 


1304.20 


36.112 




955-25 


30,906 


4o?i 


128.412 


13.2.2I 


36,223 


962.11 


31.017 


41 


128.805 


1320.25 


36.334. 


968.99 


31.128 


4i>i 


129.198 


1328.32 


36.445 


975.90 


31-238 


41X 


129.591 
129.983 


1336.40 


36.555 


982.84 


31.349 


41?^ 


1344-5' 


36.666 


989.80 


31.460 


41^ 


130.376 


1352.65 


36.777 


996.78 


31-571 


4'X 


130.769 


1360.81 


36,888 


1003,78 


31.681 


41 Ji" 


131. .61 


1369.00 


36.999 






1010.82 


31-792 


At 'A 


13'-SS4 


1377-21 


37-109 



circles: — DIAMETER, CIRCUMFERENCE, &C. 



91 



DniBeter. 


Grcum- 
ference. 


Area. 


Side of 
Equal Sauare 
(Square Root 

of Area). 


Diameter. 


Circum- 
ference. 


Area. 


Side of 
Equal Square 
(Square Root 

of Area). 


1 

42 
42'A 

42H 

4J^ 


131-947 
132.339 
132.732 
133.125 

133.518 

i33.9'o 
134.303 
134.696 


1385.44 
1393-70 
1401.98 
1410.29 
14x8.62 
1426.98 

1435-36 
1443-77 


37.220 

37.331 
37.442 

37.552 
37.663 

37.774 
37.885 
37.996 


48 
48 >^ 
48X 

48>i 
48^ 


1 50.796 
151.189 
151.582 

151-974 

152.367 
152.760 

153.153 
153.545 


1809.56 
1818.99 
1828.46 

1837.93 
1847.45 
1856.99 

1866.55 
1876.13 


42.537 
42.648 

42.759 
42.870 

42.980 

43.091 
43.202 

43.313 


•« ! 

;43H i 
j43V ; 
43H  

43>< 
.43M 
|43V 
•43J< 


135.088 

135.481 

, 135.874 
1 136.266 

136.659 
137.052 

137.445 
137.837 


1452.20 
1460.65 

1469.13 
1477-63 
1486.17 

1494.72 

1503.30 
I5II.90 


38.106 
38.217 
38.328 

38.439 

38.549 
38.660 

38.771 
38.882 


49 ^ 

49V 
49H 
49>i 
A9H 
49H 
49?^ 


153.938 

154-331 
154-723 
155.116 

155.509 
155.901 

156.294 
1 56.687 


1885.74 

1895.37 
1905.03 

1914.70 

1924.42 

1934-15 
1943.91 
1953-69 


43.423 
43.534 
43.645 
43.756 
43-867 

43.977 
44.088 

44.199 


'44 
44J< 

UH 
uH , 


138.230 
138.623 
139.015 
139.408 
139.801 
140.193 
140.586 

. 140.979 


1520.53 
1529.18 

1537.86 

1546.55 

1555.28 

1564.03 

1572.81 

1581.61 


38.993 
39.103 
39.214 
39.325 
39.436 
39.546 

39657 
39.768 


50 
50V 


1 57.080 
157.865 
158.650 
1 59.436 


1963.50 
1983.18 
2002.96 
2022.84 


44.310 

44.531 
44.753 
44.974 




160.221 
161.007 
161.792 
162.577 


2042.82 
2062.90 
2083.07 

2103.35 


45-196 

45-417 

45-639 
45.861 


4> 1 


141.372 

141.764 
142.157 

142.550 

142.942 

143.335 
143.728 

144.120 


1590.43 
1599.28 

1608.15 

1617.04 

1625.97 

1634.92 

1643.89 

1652.88 


39.879 

39.989 
40.110 

40.2 1 1 

40.322 

40.432 

40.543 
40.654 


45H 
■45X 
AiH 
45K 
45^ 
45V 


52 , 
52X 


163.363 
164.148 
164.934 
165.719 


2123.72 
2144.19 
2164.75 
2185.42 


46.082 
46.304 

46.525 
46.747 


53 , 
S3X 

S3>i 

S3J< 


166.504 
167.290 
168.075 
168.861 


2206.18 
2227.05 
2248.01 
2269.06 


46.968 
47.190 
47.411 

47.633 


46 

46H 

46X 

46X 

46>i 

46M 

46,V 

46J< 


144.513 
144-906 

145.299 

145.691 

146.084 

146.477 

146.869 

147.262 


1661.90 
1670.95 
1680.01 
1689.10 
1698.23 
1707.37 
1716.54 
1725.73 


40.765 
40.876 

40.986 

41.097 
41.208 

41.319 
41.429 

41.540 


54 , 
S4X 

S4>^ 

54JC 


169.646 
170.431 

I7I.217 
172.002 


2290.22 
2311.48 
2332.83 
2354.28 


47.854 
48.076 

48.298 

48.519 


SS 

55K 

S5^ 


172.788 

173-573 
174.358 
175-144 


2375.83 
2397-48 
2419.22 

2441.07 


48.741 
48.962 

49.184 
49-405 


47 

47H 
'47V 
, 47« 
,47>i  

i47M : 

i47V ! 
,47J< 1 

»- ____ 


147.655 
148.047 

148.440 

148.833 

149.226 

149.618 

1 50.01 1 

150.404 


1734-94 
1744.18 

1753-45 
1762.73 
1772.05 

1781.39 
1790.76 
1800.14 


41.651 
41.762 

41.873 

41.983 
42.094 

42.205 

42.316 

42.427 


56 

s(>'A 


175.929 
176.715 
177.500 
178.285 


2463.01 
2485.05 
2507.19 
2529.42 


49.627 
49.848 • 
50.070 
50.291 



92 



MATHEMATICAL TABLES. 



Diameter. 


Circum- 
ference. 


Area. 


Side of 

Equal Square 

l( Square Root 

of Area;. 


Diameter. 


Circum- 
ference. 


Area. 


Side of 
Equal Souare 
(Square Root 

of Area). 


57 ^ 
57X 
57>i 
57^ 


179.071 
179.856 
180.642 
181.427 


2551.76 

2574.19 
2596.72 

2619.35 


50.513 

50.735 
50.956 

51.178 


68 

68X 
68^ 

68^ 


213.628 
214.414 
215.199 
215.985 


3631.68 

3658UH 
3685.29 

3712.24 


60.261 
60.483 
60.704 
60.926 


58 
58X 

58^ 


182.212 
182.998 
183.783 
184.569 


2642.08 
2664.91 
2687.83 
2710.85 


51.399 
51.621 

51.842 

52.064 


69 

69X 
69X 
69^ 


216.770 

217.555 
218.341 

219.126 


3739.28 
3766.43 

3793.67 
3821.02 


61.147 
61.369 
61.591 
61.812 


59 , 
59X 

59>i 

59^ 


185.354 
186.139 

186.925 

187.710 


2733.97 
2757.19 

2780.51 

2803,92 


52.285 
52.507 
52.729 
52.950 


70 
70X 

70X 


219.912 
220.697 
221.482 
222.268 


3848.46 
3875.99 

3903.63 
3931.36 


62.034 
62.255 
62.477 
62.698 


60 

60X 

60X 
6o|<' 


188.496 
189.281 
190.066 
190.852 


2827.44 
2851.05 
2874.76 
2898.56 


53.172 
53.393 
53.615 
53.836 


71 

71X 

71^ 


223.053 
223.839 

224.624 
225.409 


3959.20 

3987.13 
4015.16 

4043.28 


62.920 
63.141 

63.363 
63.545 


61 

61X 
6i>i 

6i?< 


191.637 
192.423 
193.208 

193.993 


2922.47 
2946.47 
2970.57 
2994.77 


54.048 

54.279 
54.501 

54.723 


72 

72^ 

72 >i 

72X 


226.195 
226.980 
227.766 
228.551 

1 


4071.51 

4099.83 
4128.25 

4156.77 


63.806 
64.028 
64.249 
64.471 


62 

62X 
62 >i 

62«' 


194.779 
195.564 
196.350 

197.135 


3019.07 

3043.47 
3067.96 

3092.56 


54.944 
55.166 

55.387 
55.609 


73 
73X 

7Z)i 


229.336 
230.122 
230.907 
231.693 


4185.39 
42 14. II 
4242.92 

4271.83 


64.692 
64.914 

65.135 
65.357 


63 
63^ 
63>i 
63^ 


197.920 
198.706 
199.491 
200.277 


3117.25 
3142.04 
3166.92 

319I.9I 


55.830 
56.052 

56.273 
56.495 


74 
74V 
U% 
74H 


232.478 

233.263 

234.049 
234.834 


4300.85 

4329.95 
4359.16 
4388.47 


65.578 
65.800 
66.022 
66.243 


64 

64X 
64>i 

64l<' 


201.062 
201.847 
202.633 
203.418 


3216.99 
3242.17 
3267.46 
3292.83 


56.716 

56.938 

57.159 
57.381 


75 , 
7SX 

75>i 

7S)i 


235.620 
236.405 
237.190 
237.976 


4417.87 

4447.37 

4476.97 
4506.67 


66u^65 
66.686 
66.908 
67.129 


65 

6SX 
65 >4 

6SJ< 


204.204 
204.989 

205.774 
206.560 


3318.31 
3343.88 

3369.56 

3395.33 


57.603 

57.824 
58.046 
58.267 


76 
763^ 
76;^ 
76m: 


238.761 

239.547 
240.332 
241. 117 


4536.47 
4566.36 

4596.35 
4626.44 


67.351 1 

67.572 1 
67.794 

68.016 


66 

66X 
66;^ 

66m: 


207.345 
208.131 

208.916 

209.701 


3421.20 
3447.16 

3473.23 
3499.39 


58.489 
58.710 
58.932 
59.154 


77 
77X 
77^ 
773i 


241.903 
242.688 

243.474 
244.259 


4656.63 
4686.92 

4717.30 
4747.79 


68.237 
68.459 
68.680 
68.902 


67X 
67X 
67X 


210.487 

211.272 

' 212.058 

212.843 

1 


3525.66 
3552.01 

3578.47 
3605.03 


59.375 
59.597 
59.818 

60.040 


78 
78X 
78X 
7»H 


245.044 

245.830 
246.615 

247.401 


4778.37 
4809.05 

4839.83 
4870.70 


69.123 

69.345 
69.566 

69.788 



CIRCLES: — DIAMETER, CIRCUMFERENCE, &C 



93 



Dnuneter. 


Circuia- 
fcrence. 


Area. 


Side of 
Equal Souare 
(Square Root 

of Area). 


Diameter. 


Circum- 
ference. 


Area. 


Side of 

Equal Souare 

(Square Root 

of Area). 


79.'^ 
79«' 


248.186 
248.971 

249.757 
, 250.542 


4901.68 

4932.75 
4963.92 

4995.19 


70.009 
70.231 

70.453 
70.674 


90 

90X 
90X 

9°}i 


282.744 

283.529 
284.314 

285.099 


6361.74 
6399.12 
6432.62 
6468.16 


79.758 
79.980 
80.201 
80.423 


80 
80V 

80V 


251.328 
252.113 
252.898 
253.683 


5026.56 
5058.00 
5089.58 
5121.22 


70.896 
7I.II8 

71.339 
71.561 


91 

91X 

9i>i 

9«v: 


285.885 
286.670 
287.456 
288.242 


6503.89 
653968 

6573.56 
6611.52 


80.644 
80.866 
81.087 
81.308 


81 

8iy 

8i>i 

81 V 


254.469 

255.254 
256.040 

256.825 


5153.00 
5184.84 
5216.82 
5248.84 


71.782 
72.004 
72.225 

72.447 


92 
92X 

92H 


289.027 
289.812 
290.598 

291.383 


6647.62 
6683.80 
6720.07 
6756.40 


81.530 
81.752 

81.973 
82.195 


82 

83V 

82>i 

82V 


1 257.611 
258.396 
259.182 

, 259.967 


5281.02 
5313.28 

5345-62 
5378.04 


72.668 
72.890 

73.111 
73.333 


93 ^ 
93^ 
93X 
93V 


292.168 
292.953 

293.739 
294.524 


6792.92 
6829.48 
6866.16 
6882.92 


82.416 
82.638 
82.859 
83.081 


■83 

J83V 
83}^ 
83V 


< 260.752 

' 261.537 

262.323 

263.108 


5410.62 

5443.24 
5476.00 

5508.84 


73.554 
73.776 

73.997 
74.219 


94, 
94V 
94^ 
94V 


295.310 
296.095 
296.881 
297.666 


6939.79 
6976.72 

7013.81 

7050.92 


83.302 

83.524 
83.746 

83.968 


84 
84,V 
84^ 
84V 


1 263.894 
264.679 
265.465 
266.250 


5541.78 
5574.80 

5607.95 
5641.16 


74.440 
74.662 
74.884 
75.106 


95 , 
95X 

95 >^ 

9SH 


298.452 

299.237 
300.022 

300.807 


7088.23 
7125.56 
7163.04 
7200.56 


84189 
84.411 
84632 
84854 


85 

8;V 
85V 


267.036 
267.821 
268.606 
269.392 


5674.51 
5707.92 

5741.47 
5775.09 


75.327 

75.549 
75.770 

75.992 


96 
96V 
96'A 
96V 


301.593 
302.378 
302.164 
303.948 


7238.24 
7275.96 

7313.84 
735172 


85.077 
85.299 

85.520 
85.742 


186 

86V 
86>i 

86V 


270.177 
270.962 

271.748 

272.533 


5808.81 
5842.60 
5876.55 
5910.52 


76.213 

76.435 
76.656 

76.878 


97 
97V 
97 >i 
97^ 


304.734 
305.520 
306.306 
307.090 


7389.80 
7427.96 
7474.20 
7504.52 


85.963 
86.185 

86.407 

86.628 


87 
87V ' 

87>i ! 

87V ; 


273.319 
274-104 
274-890 
275.675 


5944.69 
5978.88 
6013.21 
6047.60 


77.099 
77.321 
77.542 
77.764 


98 
98X 

98^ 


307.876 
308.662 
309.446 
310.232 


7542.96 

7581.48 
7620.12 
7658.80 


86.850 
87.072 

87.293 
87.515 


88 

88V 
88 Ji 

88V 


276.460 

277.245 
278.031 

278.816 


6082.13 
6116.72 

6151.44 
6186.20 


77.985 
78.207 

78.428 

78.650 


99 , 
99^ 
99'A 
99^ 


3II.OI8 
311.802 
312.588 

313.374 


7697.68 
7736.60 

7775.64 
7814.76 


87.736 
87.958 
88.180 
88.401 




100 
100^ 


314.159 
315730 


7854.00 
7932.72 


88.623 
89.066 


59 
89V 

89V 


279.602 
280.387 
281.173 
281.958 


6221.15 
6256.12 
6291.25 
6326.44 


78.871 
79.093 

79.315 
79.537 


lOI 


317.301 
318.872 


8011.84 
8091.36 


89.509 
89.952 



94 



MATHEMATICAL TABLES. 



Diameter. 


Circum- 
ference. 


Area. 


Side of 

Equal Square 

^Square Root 

of Area). 


Diameter. 


Circum- 
ference. 


Area. 


Side of 

Equal Souare 

(Square Root 

of Area). 


1 02 
I02X 


320.442 
322.014 


8171.28 
8251.60 


90.395 
90.838 


112 
II2X 


351.858 
353.430 


9852.03 
9940.20 


99.258 
99.701 


103 
IO3X 


323.584 
325.154 


8332.29 
8413.40 


91.282 
91.725 


113 
II3>^ 


355.000 
356.570 


10028.75 

loi 17.68 


100.144 
100.587 


104 

io4}4 


326.726 
328.296 


8494.87 
8576.76 


92.168 
92.61 1 


114 


358,142 
359.712 


10207.03 
10296.76 


101.03 1 
101.474 


105 


329.867 
331.438 


8659.01 
8741.68 


93.054 
93.497 


115 . 

"5>^ 


361.283 
362.854 


10386.89 
10477.40 


IOI.917 
102.360 


106 


333.009 
334.580 


8824,73 
8908.20 


93.940 
94.383 


116 
116;^ 


364.425 
365.996 


10568.32 
10659.64 


102.803 
103.247 


107 
107 >^ 


336.150 
337.722 


8992.02 
9076.24 


94.826 
95.269 


117 


367.566 
369.138 


10751.32 
10843.40 


103.690 
104.133 


108 
108K 


339.292 
340.862 


9160.88 
9245.92 


96.156 


118 

ii8>^ , 


370.708 
372.278 


10935.88 
1 1028.76 


104.576 
105.019 


109 
109;^ 


342.434 
344.004 


9331.32 
9417.12 


96.599 
97.042 


119 
119^ 


373.849 
375.420 


1 1 122.02 
1 121 5.68 


105.463 
105.906 


no 


345.575 
347.146 


9503.32 
9589.92 


97.485 
97.928 


120 


376.991 


M 309.73 


106.350 


III 

MiX 


348.717 
350.288 


9676.89 
9764.28 


98.371 
98.815 




LENGTHS OF CIRCULAR ARCS. 



95 



TABLE No. v.— LENGTHS OF CIRCULAR ARCS FROM 
l^ TO l8o^ GIVEN, THE DEGREES. 

(Radius = i.) 



Degrees. 


Length. 


' Degrees. 

1 


Length. 


Degrees. 


Length. 


Degrees. 


Length. 


I 


.0174 


40 


.6981 


79 


1.3788 


117 


2.0420 


2 


.0349 


41 


.7156 


80 


1-3963 


118 


2.0595 


3 


-0524 


42 


.7330 


119 


2.0769 


4 


.0698 


43 


.7505 


81 




1-4137 






5 


.0873 


44 


.7679 


82 


I.4312 


120 


2.0944 


6 


.0147 


45 


.7854 


83 


1.4486 


121 


2.II18 


7 


.0222 


46 


.8028 


84 


1. 4661 


122 


2.1293 


8 


.0396 


47 


.8203 


85 


1.4835 


123 


2.1467 


9 


-0571 


48 


.8377 


86 


I.50IO 


124 


2.1642 






49 


.8552 


87 


1.5^84 


125 


2.I8I7 


lO 


.1745 


^ • 


w %/ 


88 


1-5359 


126 


2.1991 


II 


.1920 


50 


.8727 


89 


1-5533 


127 


2.2166 


12 


.2094 


SI 


.8901 




^% 


128 


2.2304 


13 


.2269 


5» 


.9076 


90 


1.5708 


129 


2.2515 


14 


.2443 


S3 


.9250 


91 


L5882 






15 


.2618 


54 


.9424 


92 


1.6057 


130 


2.2689 


16 


.2792 


SS 


•9599 


93 


L6231 


131 


2.2864 


17 


.2967 


S6 


•9774 


94 


1.6406 


132 


2.3038 


18 


-314I 


57 


.9948 


95 


L6581 


133 


2.3213 


19 


'ZZ^^ 


S8 


1.0123 


96 


1.6755 


134 


2.3387 


^ 




59 


1.0297 


97 


1.6930 


135 


2.3562 


ao 


.3491 






98 


L7I04 


136 


2.3736 


21 


.3665 


60 


1.0472 


99 


1.7279 


137 


2.391 1 


22 


.3840 


61 


1.0646 






138 


2.4085 


23 


.4014 


62 


1. 0821 


100 


1.7453 


139 


2.4260 


24 


.4189 


63 


10995 


lot 


1. 2628 


1 




25 


•4363 


64 


1.1170 


102 


L7802 


140 


2.4435 


26 


.4538 


65 


I-I345 


103 


1.7977 


141 


2.4609 


27 


.4712 


66 


1.1519 


104 


1.8151 


142 


2.4784 


28 


.4887 


67 


1. 1694 


105 


1.8326 


143 


2.4958 


29 


.5061 


68 


1. 1868 


106 


1.8500 


144 


2.5133 


^ 




69 


1.2043 


107 


1.8675 


145 


2.5307 


30 


.5236 






108 


L8849 


C46 


2.5482 


31 


.54IC) 


70 


1. 2217 


109 


1.9024 


147 


2.5656 


32 


.5585 


71 


1.2392 






148 


2.5831 


33 


.5759 


72 


1.2566 


no 


1.9199 


149 


2.6005 


34 


.5934 


73 


1-2741 


III 


1-9373 






. 35 


.6109 , 


74 


L2915 


112 


1.9548 


150 


2.6180 


36 


.6283 


75 


1.3090 


113 


1.9722 


151 


2.6354 


37 


.645S ! 


76 


1.3264 


114 


1.9897 


152 


2.6529 


38 


.66:^2 


77 


1.3439 


115 


2.0071 


153 


2.6703 


1 39 


.6807 


78 


1.3613 


ii6 


2.0246 


154 


2.6878 









i:"^i. 



96 



MATHEMATICAL TABLES. 



Degrees. 


Length. 


Degrees. 


Length. 


Degrees. 


Length. 


Degrees. 


Length. 


155 


2.7053 


161 


2.8100 


168 


2.9321 


174 


3.0369 


156 


2.7227 


162 


2.8274 


169 


2.9496 


175 


30543 


157 


2.7402 


163 


2.8449 






176 


3.0718 


158 


2.7576 


164 


2.8623 


170 


2.9670 


177 


3.0892 


159 


2.7751 


165 


2.8798 


171 


2.9845 


178 


3.1067 






166 


2.8972 


172 


3.0020 


179 


3. 1 241 


160 


2.7925 


167 


2.9147 


173 


30194 


180 


3.I416 



L 



LENGTHS OP CIRCULAR ARCS. 



97 



TABLE No. VI.— LENGTHS OF CIRCULAR ARCS, UP TO A 

SEMICIRCLE. GIVEN, THE HEIGHT. 

(Chord = I.) 



Height. 


Length. 


HdghL 


Length. 


Height. 


Length. 


Height. 


Length. 


.100 


1.02645 


.140 


I.05147 


.180 


1.08428 


.220 


1. 1 2445 


.101 


1.02698 


.141 


1.05220 


.181 


I.08519 


.221 


I.I2556 


-I02 


1.02752 


.142 


1.05293 


.182 


I.0861I 


.222 


1. 12663 


.103 


1.02806 


.143 


1.05367 


.183 


1.08704 


.223 


1.12774 


.104 


1.02860 


.144 


I.0544I 


.184 


1.08797 


.224 


1. 12885 


-105 


1. 02914 


.145 


I.05516 


.185 


1.08890 


.225 


1. 12997 


.106 


1.02970 


.146 


I.0559I 


.186 


1.08984 


.226 


1.13108 


.107 


1.03026 


.147 


1.05667 


.187 


1.09079 


.227 


I.I3219 


.108 


1.03082 


.148 


1.05743 


.188 


I.O9174 


.22-8 


1.13331 


.109 


1-03139 


.149 


I.05819 


.189 


1.09269 


.229 


1. 13444 


.110 


I.O3I96 


.150 


1.05896 


.190 


1.09365 


.230 


I.I3557 


.III 


1.03254 


.151 


1.05973 


.191 


I.O9461 


.231 


I.I3671 


.112 


I.O3312 


.152 


1.0605 1 


.192 


1.09557 


.232 


I.I3786 


."3 


I.O3371 


.153 


1. 06130 


.193 


1.09654 


.233 


1-13903 


.114 


1.03430 


.154 


1.06209 


.194 


1.09752 


•234 


1. 14020 


.1^5 


1.03490 


•155 


1.06288 


.195 


1.09850 


.235 


I.I4136 


.116 


I.O3551 


.156 


1.06368 


.196 


1.09949 


.236 


I.I4247 


.117 


I.O361I 


.157 


1.06449 


.197 


1. 10048 


.237 


1-14363 


.118 


1.03672 


.158 


1.06530 


.198 


I.IOI47 


.238 


1. 14480 


.119 


1.03734 


.159 


I.0661I 


.199 


1. 10247 


.239 


r. 14597 


.120 


1.03797 


.160 


1.06693 


.200 


1. 10348 


.240 


1.14714 


.121 


1.03860 


.161 


1.06775 


.201 


1. 10447 


.241 


1-14831 


.122 


1.03923 


.162 


1.06858 


.202 


I.IO548 


.242 


1. 14949 


.123 


1.03987 


.163 


I.0694I 


.203 


1. 10650 


.243 


1. 15067 


.124 


1.0405 1 


.164 


1.07025 


.204 


I.IO752 


.244 


1.15186 


•125 


I.04I16 


.165 


1. 07109 


.205 


I.I0855 


.245 


1.15308 


.126 


I.04181 


.166 


I.07194 


.206 


I.IO958 


.246 


1.15429 


.127 


1.04247 


.167 


1.07279 


.207 


I.II062 


.247 


1.15549 


.128 


1.043 13 


.168 


1.07365 


.208 


I.II165 


.248 


1.15670 


.129 


1.04380 


.169 


I.07451 


.209 


I.II269 


.249 


1.15791 


.130 


1.04447 


.170 


1.07537 


.210 


I.II374 


.250 


1.15912 


-131 


I.O4515 


.171 


1.07624 


.211 


I.II479 


.251 


1. 16033 


.132 


1.04584 


.172 


I.O771I 


.212 


I.II584 


.252 


1.16157 


'^33 


1.04652 


.173 


1.07799 


.213 


I.I1692 


•253 


1. 16279 


•134 


1.04722 


.174 


1.07888 


.214 


I.II796 


.254 


1. 16402 


-135 


1,04792 


.175 


1.07977 


.215 


1. 1 1904 


.255 


1. 16526 


.136 


1.04862 


.176 


1.08066 


.216 


I.I20II 


.256 


1. 16649 


•^37 


1.04932 


.177 


I.08156 


.217 


I.I2II8 


.257 


1.16774 


.13S 
•^39 / 


1.05003 


.178 


1.08246 


.218 


1. 12225 


.258 


1. 16899 


105075 \ 


.179 


1.08337 


.219 


I.I2334 


.259 


1. 17024 



98 



MATHEMATICAL TABLES. 



Height, 



Length. 



.260 

.261 
.262 
.263 
.264 
.265 
.266 
.267 
.268 
.269 

.270 
.271 
.272 

.273 
.274 

.275 
.276 

.277 

.278 

.279 

.280 

.281 
.282 
.283 
.284 
.285 
.286 
.287 
.288 
.289 

.290 
.291 
.292 

•293 
.294 

•295 
.296 

.297 

.298 

.299 

.300 

.301 
.302 

•303 
•304 

•305 
.306 



.17150 

.17275 
.17401 

.17527 

.17655 
.17784 

.17912 

.18040 

.18162 

.18294 

.18428 

•18557 
.18688 

.18819 

.18969 

.19682 

.19214 

.19345 

•19477 
.19610 

.19743 
.19887 

.20011 

.20146 

.20282 

.20419 

.20558 

.20696 

.20828 

.20967 

.21202 
.21239 
.21381 
.21520 
.21658 
.21794 
.21926 
.22061 
.22203 
.22347 

.22495 
.22635 
.22776 
.22918 
.23061 
.23205 

•23349 



Height 



Length. 



.307 
.308 

.309 
.310 

.311 
.312 

.313 
.314 

.315 
.316 

.317 
.318 

.319 

.320 
.321 
.322 

.323 

.324 

.325 
.326 

•327 
.328 

•329 
.330 

•332 
'333 
.334 
.335 
•336 

.337 
.338 
.339 

.340 

•341 
.342 

.343 
.344 

•345 
.346 

•347 
.348 

.349 

.350 
.351 
.352 
.353 



•23494 
.23636 

.27780 

•23925 
.24070 

.24216 

.24360 

.24506 

.24654 

.24801 

.24946 

.25095 
.25243 

.25391 

•25539 
.25686 

.25836 

.25987 

•26137 

.26286 

.26437 

.26588 

.26740 

.26892 
.27044 
.27196 

.27349 
.27502 

.27656 

.27810 

.27864 

.28118 

.28273 

.28428 
.28583 
.28739 
.28895 
.29052 
.29209 
.29366 

•29523 
.29681 

•29839 

.29997 
.30156 
.30315 
.30474 



Height. 



•354 
•355 
•356 
•357 
•358 
.359 
.360 
.361 
.362 

'3^3 

.364 

.365 
.366 

.367 
.368 

.369 

.370 

.371 
.372 

.373 
.374 

.375 
.376 

.377 
.378 

.379 

.380 

.381 
.382 

.383 
.384 

.385 
.386 

.387 
.388 

.389 

.390 
.391 
.392 

•393 
•394 
.395 
.396 
.397 
.398 
•399 
.400 



Length. 



•30634 
•30794 
•30954 
.31115 
.31276 

•31437 

•31599 
.31761 

•3^923 
.32086 

•32249 

.32413 

•32577 

.32741 

•32905 
.33069 

.33234 
.33399 
.33564 
.33730 
.33896 
.34063 
.34229 

.34396 
.34563 
.34731 

.34899 
.35068 

•35237 
.35406 

•35575 
•35744 

.35914 
.36084 

.36254 
.36425 

.36596 
.36767 

•36939 
.37111 

.37283 

.37455 
.37628 

.37801 

.37974 
.38148 

.38322 



Height. 

.401 
.402 
.403 
.404 
.405 
.406 

.407 
.408 

.409 

.410 
.411 
.412 

.413 
.414 

.415 
.416 

.417 

.418 

.419 

.420 

.421 
.422 

•423 
.424 

.425 
.426 

.427 
.428 

.429 

.430 
.431 
.432 

.433 
.434 
.435 
.436 

•437 
.438 

•439 
.440 
.441 
.442 

.443 
.444 

•445 
.446 

.447 



Length. 

[.38496 

.38671 
.38846 

•39021 
.39196 

39372 
•30548 

•39724 
.39900 

.40077 
.40254 
.40432 
.40610 
.40788 
.409^6 

.41145 
■41324 

■41503 
.41682 

.41861 
.42041 
.42222 
.42402 

•42583 
.42764 

•42945 
•43127 
•43309 
.43491 

•43673 
•43856 

•44039 
.44222 

.44405 
.44589 

•44773 
•44957 
■45142 
■45327 

•455" 

•45697 

■45883 
.46069 

•46255 

.46441 

.46628 

.46815 



LENGTHS OF CIRCULAR ARCS. 



99 



Hdght. 



.448 

•449 

.450 

.451 
•452 

•453 
•454 
•455 
•45 <^ 
•457 
.458 
•459 



JLength. 



1.47002 
1. 47189 

1.47377 

1.47565 

1.47753 

1.47942 
I.48131 

1.48320 

1.48509 

1.48699 

1.48889 

1.49079 



.460 1.49269 



Height. 



.461 
.462 

.463 
.464 

.465 
.466 

.467 

.468 

.469 

.470 
.471 

.472 

.473 
.474 



Length. 



.49460 
.49651 
.49842 

•50033 
.50224 

.50416 

.50608 

.50800 

.50992 

•51185 
•51378 
■5157I 
.51764 
•51958 



Height 



Length. 



.475 
.476 

.477 
.478 
.479 

480 

.481 
.482 

.483 
.484 

.485 
.486 

.487 
.488 



52^52 
52346 
52541 
52736 

52931 

53126 
53322 

53518 

53714 

53910 
54106 

54302 

54499 
54696 



Height 
.489 

.490 
.491 

.492 

•493 

•494 

•495 
496 

.497 
.498 
499 
.500 



Length. 

1-54893 

55090 
.55288 

.55486 

.55685 

•55854 
.56083 

.56282 

.56481 

.56680 

.56879 

-57079 



MATHEMATICAL TABLES. 



TABLE No. VIL— AREAS OF CIRCULAR SEGMENTS, UP TO A 

SEMICIRCLE. 



AREAS OF CIRCULAR SEGMENTS. 



101 



, HdghL 

1 

1 


Area. 


Height. 


Area. 


Height. 


Area. 


Height. 


1 

1 

Area. 


'.X57 


.07892 


.203 


.11423 


.249 ; 


.15268 


•295 


.19360 


.158 


.07965 ' 


.204 


.11504 


.250 


• IS3SS 


.296 


. -19451 


•159 


.08038 


.205 


.11584 


•251 
.252 


.15442 
.15528 


.297 


•19543 


.160 


.08111 


.206 


.11665 


.298 


.19634 


.161 
.162 


.08185 
.08258 


.207 
.208 

>k ^v ^^ 


.11746 
.11827 
.11908 


.253 
.254 


.15615 
.15702 


.299 
.300 


•19725 
.19817 


.163 


.08332 


.209 


.255 


.15789 


.301 


.19908 


.164 


.08406 


.210 


.11990 


.256 


.15876 


.302 


.20000 


.165 


.08480 


.211 


I2071 


.257 


.15964 


•303 


.20092 


. .166 


.08554 


.212 


I2153 


.258 


.16051 


.304 


.20184 


.167 


.08629 


.213 


12235 


•259 


.16139 


•305 


.20276 


.168 

1 ^ 


.08704 


.214 


I2317 


.260 


.16226 


.306 


.20368 


.169 


.08778 


.215 . 


.12399 


.261 


.16314 


•307 


.20460 


.170 


.08854 


.216 


1 248 1 


.262 


.16402 


.308 


•20553 


.171 


.08929 


.217 


12563 


.263 


.16490 


•309 


.20645 


.172 


.09004 


.218 


12646 


.264 


.16578 


.310 


.20738 


.173 


.09080 


.219 


12729 


.265 


16666 


%0 

•3" 


.20830 


.174 


•09155 


.220 


I281I 


.266 


1675s 


.312 


.20923 


•175 


.09231 


.221 


12894 


.267 


16843 


•313 


.21015 


.176 


.09307 


.222 


12977 


.268 


16932 


•314 


.21108 


•'^2 


-09383 


.223 


13060 


.269 


17020 


•315 


.21201 


.178 


.09460 


.224 


I3144 


.270 


.17109 


.316 


.21294 


.179 


.09537 


.225 


13227 


.271 


.17198 


.317 


.21387 


.180 


.09613 


.226 


133" 


.272 


.17287 


.318 


.21480 


.181 


.09690 


.227 


13395 


.273 


.17376 


•319 


•21573 . 


.182 


.09767 


.228 


13478 


.274 


.17465 


.320 


.21667 


.183 


.09845 


.229 


13562 


.275 


17554 


•J 
•321 


.21760 


.184 


.09922 


.230 


13646 


.276 


.17644 


.322 


•21853 


i-'f5 


.09200 


.231 


I373I 


.277 


'I7733 


•323 


.21947 


.186 


.10077 


.232 


I3815 


.278 


.17823 


.324 


.22040 


.187 

€% 


.10153 


.233  


13899 


.279 


.17912 


.325 


.22134 


.188 


.10233 


.234 


13984 


.280 


.18002 


.326 


.22228 


.189 


.10317 


.235 


14069 


.281 


.18092 


.327 


.22322 


.190 


.10390 


.236 


I4I54 


.282 


.18182 


.328 


.22415 


,.191 


.10469 


.237 


14239 


.283 


.18272 


•329 


.22509 


.192 


.10547 


.238 


14324 


.284 


.18362 


•330 

•332 

•333 
.334 
•335 


.22603 
.22697 
.22792 
.22886 
.22980 
.23074 


•193 
.194 

.195 

.196 

.197 

.198 

.» ^ _ 


.10626 
.10705 
.10784 
.10864 

.10943 


•239 • 

.240 
.241 
.242 

.243 


14409 

14494 
14580 
14665 

14752 


.285 
.286 
.287 
.288 
.289 


.18452 

.18542 
.18633 
.18723 
.18814 


.11023 


.244 


•14837 


.290 


.18905 


.336 


.23169 


1 .199 


.11102 


.245 


14923 


.291 


.18996 


•337 


.23263 


.200 


.11182 


.246 


.15009 


.292 


.19086 


•338 


•23358 


.201 


.11262 


.247 


.15096 


•293 


.19177 


•339 


•23453 


.202 


.11343 


.248 


.15182 


•294 


.19268 


.340 


.^3547 



SINES, COSINES, &C OF ANGLES. 



103 



TABLE No. VIIL— SINES, COSINES, TANGENTS, COTANGENTS, 
SECANTS, AND COSECANTS OF ANGLES FROM 0° to 90°- 

Advancing by 10' or one-sixth of a degree. (Radius =i.) 



1 Angle. 


Sine. 


Cosecant. 


Tangent 


Cotangent 


Secant 

• 


Cosine. 






0' 0' 


.000000 


Infinite. 


.000000 


Infinite. 


1. 00000 ] 


[.000000 


90° 0' 




10 


.002909 


343-77516 


.002909 


343.77371 


1. 00000 


.999996 


50 




20 


.005818 


171.88831 


.005818 


171.88540 


1.00002 


.999983 


40 




1 30 


.008727 


114.59301 


.008727 


114.58865 


1.00004 


.999962 


30 




40 


.011635 


85-945609 


.011636 


85.939791 


1.00007 


.999932 


20 




50 


.014544 


68.757360 


.014545 


68.750087 


1. 000 1 1 


.999894 


10 




I 


.017452 


57.298688 


.017455 


57.289962 


I.OOOI5 


.999848 


89 




10 


.020361 


49.114062 


.020365 


49.103881 


1.0002 1 


.999793 


50 




20 


.023269 


42.975713 


.023275 


42.964077 


1.00027 


.999729 


40 




1 30 


.026177 


38.201550 


.026186 


38.188459 


1.00034 


•999657 


30 




40 


.029085 


34.382316 


.029097 


34.367771 


1.00042 


.999577 


20 




' 50 


.031992 


31.257577 


.032009 


31.241577 


1.0005 1 


.999488 


10 




2 


.034899 


28.653708 


.034921 


28.636253 


1.0006 1 


.999391 


88 




' 10 


.037806 


26.450510 


•037834 


26.431600 


1.00072 


.999285 


50 




20 


.040713 


24.562123 


.040747 


24.541758 


1.00083 


.999171 


40 




30 


.043619 


22.925586 


.043661 


22.903766 


1.00095 


.999048 


30 




40 


.046525 


21.493676 


.046576 


21.470401 


1. 00108 


.998917 


20 




SO 


.049431 


20.230284 


.049491 


20.205553 


1. 00122 


.998778 


10 




3 P 


.052336 


19.107323 


.052408 


19.081 137 


I.OOI37 


.998630 


87 




10 


•055241 


18.102619 


.055325 


18.074977 


I.OOI53 


.998473 


50 




20 


.058145 


17.198434 


.058243 


17-169337 


1. 00169 


.998308 


40 




30 


.061049 


16.380408 


.061163 


16.349855 


1. 00187 


•998135 


30 




40 


.063952 


15.636793 


.064083 


15.604784 


1.00205 


.997357 


20 




50 


.066854 


14.957882 


.067004 


14.924417 


1.00224 


•997763 


10 




4 


.069756 


14.335587 


.069927 


14.300666 


1.00244 


.997564 


86 




10 


.072658 


i3.763"5 


.072851 


13.726738 


1.00265 


.997357 


50 




20 


•075559 


13.234717 


.075776 


13.196888 


1.00287 


.997141 


40 




30 


.078459 


12.745495 


.078702 


12.706205 


1.00309 


.996917 


30 




40 


.081359 


12.291252 


.081629 


12.250505 


1.00333 


.996685 


20 




50 


.084258 


11.868370 


.084558 


II. 826167 


1.00357 


.996444 


10 




5 


.087156 


11-473713 


.087489 


11.430052 


1.00382 


.996195 


85 




10 


.090053 


11.104549 


.090421 


11.059431 


1.00408 


.995937 


50 




20 


.092950 


10.758488 


.093354 


10.711913 


1.00435 


.995671 


40 




30 


.095846 


10.433431 


.096289 


10.385397 


1.00463 


.995396 


30 




40 


.098741 


10.127522 


.099226 


10.078031 


1.0049 1 


•995113 


20 


 


50 


.101635 


9.8391227 


.102164 


9.7881732 


1.005 2 1 


.994822 


10 

Angle. 






Cosine. 


Secant 


Cotangent 


Tangent 


Cosecant 


Sme. 





I04 



MATHEMATICAL TABLES. 



Angle. 


Sine. 


Cosecant. 


Tangent. 


Cotangent 


Secant 


Cosine. 




6° o' 


.104528 


9.5667722 


.105104 


95143645 


1.0055 X 


•994522 


84° 0' 


lO 


.107421 


9.3091699 


.X 08046 


92553035 


X. 00582 


.9942x4 


50 


20 


.110313 


9.0651512 


.X 10990 


9.OO9826X 


X.O0614 


•993897 


40 


30 


.113203 


8.8336715 


.113936 


8.7768874 


1.00647 


.993572 


30 


40 


.116093 


8.613790I 


.XX6883 


8.5555468 


i.oo68x 


•993238 


20 


50 


.118982 


8.4045586 


•119833 


8.3449558 


1.007x5 


.992896 


10 


7 


.121869 


8.2055090 


.122785 


8.x 443464 


1.0075X 


.992546 


83 


10 


.124756 


8.0156450 


.125738 


7.9530224 


X.00787 


.992x87 


50 


20 


.127642 


7-8344335 


.X 28694 


7.7703506 


1.00825 


.99x820 


40 


30 


.130526 


7.6612976 


.131653 


7.5957541 


X. 00863 


•991445 


30 


40 


•I334IO 


7.4957100 


.134613 


7.4287064 


X.00902 


.99106X 


20 


50 


.136292 


7^3371909 


.137576 


7.2687255 


X.00942 


.990669 


10 


8 


•I39173 


7.1852965 


.140541 


7^ii53697 


X. 00983 


.990268 


82 


10 


•142053 


7.0396220 


•143508 


6.9682335 


X. 0x024 


.989859 


50 


20 


.144932 


6.8997942 


.146478 


6.8269437 


X. 0x067 


•989442 


40 


30 


.147809 


6.7654691 


.X 4945 1 


6.691x562 


x.oxxxx 


.989016 


30 


40 


.150686 


6.6363293 


.152426 


6.5605538 


I-OII55 


.988582 


20 


50 


•I53561 


6^5120812 


.155404 


6.4348428 


1.0x200 


.988x39 


10 


9 


.156434 


6.3924532 


.158384 


6.3137515 


1.01247 


.987688 


81 


10 


•159307 


6.2771933 


.161368 


6.X970279 


X. 01294 


.987229 


50 


20 


.162178 


6.1660674 


•164354 


6.0844381 


1.0x432 


.986762 


40 


30 


.165048 


6.0588980 


.167343 


59757644 


X.OI39I 


.986286 


30 


40 


.167916 


59553625 


•170334 


5.8708042 


1.0x440 


.98580X 


20 


50 


.170783 


5.8553921 


.173329 


5.7693688 


X.OI49I 


•985309 


XO 


10 


.173648 


5-7587705 


.176327 


5.6712818 


1-01543 


.984808 


80 


10 


.176512 


5-6653331 


.179328 


5-5763786 


1^01595 


.984298 


* 50 


20 


•179375 


5-5749258 


.182332 


5.4845052 


X. 0x649 


.983781 


40 


30 


.182236 


5.4874043 


•185339 


5.3955172 


X.OI703 


•983255 


30 


40 


.185095 


5.4026333 


.188359 


5^3092793 


1.0x758 


.98272X 


20 


50 


•187953 


5.3204860 


.191363 


5.2256647 


x.ox8x5 


.982x78 


10 


II 


.190809 


5.2408431 


.194380 


5.1445540 


X. 0x872 


.98x627 


79 


10 


.193664 


5.1635924 


.X9740X 


5-0658352 


X. 01930 


.981068 


50 


20 


.196517 


5.0886284 


.200425 


4.9894027 


1.0x989 


.980500 


40 


30 


.199368 


5.OI58317 


.203452 


49151570 


1.02049 


•979925 


30 


40 


.202218 


4.9451687 


.206483 


4.8430045 


1.02110 


•979341 


20 


50 


.205065 


4.8764907 


.209518 


4.7728568 


1.0217X 


.978748 


10 


12 


.207912 


4.8097343 


.212557 


4.7046301 


1.02234 


.978148 


78 


10 


.210756 


4.7448206 


•215599 


4.6382457 


1.02298 


•977539 


SO 


20 


•213599 


4.6816748 


.218645 


4-5736287 


1.02362 


.976921 


40 


30 


.216440 


4.6202263 


.22x695 


4.5x07085 


1.02428 


.976296 


30 


40 


.219279 


4.5604080 


.224748 


4.4494x81 


1.02494 


.975662 


20 


50 


.2221X6 


4-502x565 


.227806 


4,3896940 


1.02562 


.975020 


10 


1 


Coune. 


Secant 


Cotangent 


Tangent 


Cosecant 


Sine. 


Angle. 



SINES, COSINES, &C OF ANGLES. 



105 



Angle. 


Sine. 


Cosecant. 


Tangent. 


Cotangent 


Secant. 


Cosine. 




13° 0' 


.224951 


4.44541 15 


.230868 


4.3314759 


1.02630 


•974370 


77" 0' 


10 


.227784 


4.3901 158 


•233934 


4.274706*6 


1.02700 


.973712 


50 


20 


.230616 


4.3362150 


.237004 


4.2193318 


1.02770 


•973045 


40 


30 


•233445 


4-2836576 


.240079 


4.1652998 


1.02842 


.972370 


30 


40 


.236273 


4^2323943 


.243158 


4.II25614 


1. 02914 


.971687 


20 


50 


.239098 


4,1823785 


.246241 


4.0610700 


1.02987 


•970995 


10 


14 


.241922 


4.1335655 


.249328 


4,0107809 


I.O3061 


.970296 


76 


1 10 


•244743 


4.0859130 


.252420 


3.9616518 


IO3137 


.969588 


50 


' 20 


•247563 


4.0393804 


•255517 


3.9136420 


I.O3213 


.968872 


40 


30 


.250380 


3.9939292 


.258618 


3.866713I 


1.03290 


.968148 


30 


40 


•253195 


3.9495224 


.261723 


3.8208281 


1.03363 


.967415 


20 


50 


.256008 


3.9061250 


.264834 


3.7759519 


1.03447 


.966675 


10 


15 


.258819 


3.8637033 


•267949 


3.7320508 


1.03528 


.965926 


75 


10 


.261628 


3.8222251 


.271069 


3.6890927 


1.03609 


.965169 


50 


20 


.264434 


3.7816596 


.274195 


3.6470467 


I.O369I 


.964404 


40 


30 


.267238 


3^74i9775 


•277325 


3.6058835 


1.03774 


.963630 


30 


40 


.^70040 


3.7031506 


.280460 


3.5655749 


1.03858 


.962849 


20 


50 


.272840 


3.6651518 


.283600 


3.5260938 


i.o39'44 


.962059 


10 


16 


•275637 


3.6279553 


.286745 


3.4874144 


1.04030 


.961262 


74 


10 


.278432 


3-5915363 


.289896 


3.4495120 


1.04117 


.960456 


50 


20 


.281225 


3-5558710 


.293052 


3.4123626 


1.04206 


.959642 


40 


, 30 


.284015 


3.5209365 


.296214 


3.3759434 


1.04295 


.958820 


30 


40 


.286803 


3.48671 10 


.299380 


3.3402326 


1.04385 


•957990 


20 


50 


.289589 


3^4531735 


•302553 


3.3052091 


1.04477 


.957151 


10 


17 


.292372 


3.4203036 


.305731 


3.2708526 


1.04569 


•956305 


73 


10 


•295152 


3.3880820 


.308914 


3.2371438 


1.04663 


.955450 


50 


20 


.297930 


33564900 


.312104 


3.2040638 


1.04757 


•954588 


40 


30 


.300706 


3^3255095 


.315299 


3.1715948 


1.04853 


•953717 


30 


40 


•303479 


3.2951234 


.318500 


3-1397194 


1.04950 


•952838 


20 


50 


.306249 


3.2653149 


.321707 


3.I0842IO 


1.05047 


•951951 


10 


18 


.309017 


3.2360680 


.324920 


3-0776835 


1.05146 


•951057 


72 


10 


.311782 


3.2073673 


.328139 


3.0474915 


1.05246 


•950154 


50 


20 


•314545 


3.1791978 


•331364 


3.OI783OI 


1-05347 


.949243 


40 


30 


•317305 


3-1515453 


.334595 


2.9886850 


1.05449 


.948324 


30 


40 


.320062 


3.1243959 


.337833 


2.9600422 


1.05552 


.947397 


20 


50 


.322816 


30977363 


•341077 


2.9318885 


1.05657 


.946462 


10 


19 


•325568 


3^o7i5535 


.344328 


2.9042109 


1.05762 


•945519 


71 


10 


•328317 


3^0458352 


.347585 


2.8769970 


1.05869 


.944568 


50 


20 


•331063 


3.0205693 


.350848 


2.8502349 


1.05976 


•943609 


40 


30 


•333807 


2.9957443 


.354119 


2.8239129 


1.06085 


.942641 


30 


40 


•336547 


2.9713490 


.357396 


2.7980198 


1.06195 


.941666 


20 


50 


•339285 


2.9473724 


.360680 


2.7725448 


1.06306 


.940684 


10 




Cosine. 


Secant. 


Cotangent. 


Tangent 


Cosecant. 


i 
Sine. 


Angle. 



io6 



MATHEMATICAL TABLES. 



Angle. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 




20** O' 


.342020 


2.9238044 


•363970 


2.7474774 


1. 06418 


•939693 


70° 0' 


lO 


•344752 


2.9006346 


.367268 


2.7228076 


I.0653I 


.938694 


50 


20 


.347481 


2.8778532 


•370573 


2.6985254 


1.06645 


.937687 


40 


30 


.350207 


2.8554510 


•373885 


2.6746215 


1. 06761 


.936672 


30 


40 


•352931 


2.8334185 


•377204 


2.6510867 


1.06878 


.935650 


20 


50 


•355651 


2.8117471 


.380530 


2.627912I 


1.06995 


•934619 


10 


21 


.358368 


2.7904281 


.383864 


2.6050891 


I.07115 


•933580 


69 


10 


.361082 


2.7694532 


.387205 


2.5826094 


^•07235 


•932534 


50 


20 


•363793 


2.7488144 


.390554 


2.5604649 


1.07356 


.931480 


40 


30 


.366501 


2.7285038 


.393911 


2.5386479 


1.07479 


.930418 


30 


40 


.369206 


2.7085139 


.397275 


2.5171507 


1.07602 


.929348 


20 


50 


.371908 


2.6888374 


.400647 


2.4959661 


1.07727 


.928270 


10 


22 


.374607 


2.6694672 


.404026 


2.4750869 


107853 


.927184 


68 


10 


■377302 


2.6503962 


.407414 


2.4545061 


1. 07981 


.926090 


50 


20 


•379994 


2.6316180 


.410810 


2.4342172 


1. 08109 


.924989 


40 


30 


.382683 


2.6131259 


.414214 


2.4142136 


1.08239 


.923880 


30 


40 


•385369 


2.5949137 


.417626 


2.3944889 


1.08370 


.922762 


20 


SO 


.388052 


2.5769753 


.421046 


2.3750372 


1.08503 


.921638 


10 


23 


.390731 


2.5593047 


.424475 


2.3558524 


1.08636 


•920505 


67 


10 


•393407 


2.5418961 


.427912 


2.3369287 


I.08771 


.919364 


50 


20 


.396080 


2.5247440 


•431358 


2.3182606 


1.08907 


.918216 


40 


30 


•398749 


2.5078428 


.434812 


2.2998425 


1.09044 


.917060 


30 


40 


.401415 


2.491 1874 


.438276 


2.2816693 


I.O9183 


.915896 


20 


50 


.404078 


2.4747726 


.441748 


2.2637357 


1.09323 


.914725 


10 


24 


.406737 


2.4585933 


•445229 


2.2460368 


1.09464 


•913545 


66 


10 


.409392 


2.4426448 


.448719 


2.2285676 


1.09606 


•912358 


50 


20 


.412045 


2.4269222 


.452218 


2.2II3234 


1.09750 


.911164 


40 


30 


.414693 


2.4II42IO 


.455726 


2.1942997 


1.09895 


.909961 


30 


40 


.417338 


2.3961367 


.459244 


2.1774920 


I.IOO4I 


.908751 


20 


SO 


.419980 


2.3810650 


.462771 


2.1608958 


I.IO189 


.907533 


10 


25 


.422618 


2.3662016 


.466308 


2.1445069 


I.IO338 


.906308 


65 


10 


.425253 


2.3515424 


•469854 


2.I283213 


1. 10488 


•905075 


50 


20 


.427884 


2.3370833 


•473410 


2. 1 1 23348 


1. 10640 


•903834 


40 


30 


•4305 1 1 


2.3228205 


•476976 


2.0965436 


I.IO793 


•902585 


30 


40 


•433^35 


2.3087501 


.480551 


2.0809438 


1. 10947 


.901329 


20 


so 


•435755 


2.2948685 


.484137 


2.0655318 


I.III03 


.900065 


10 


26 


•438371 


2.281172O 


.487733 


2.0503038 


I.II260 


.898794 


64 


10 


.440984 


2.2676571 


.491339 


2.0352565 


I.II419 


.897515 


50 


20 


•443593 


2.2543204 


.494955 


2.0203862 


I.II579 


.896229 


40 


30 


.446198 


2.24II585 


.498582 


2.0056897 


1. 1 1 740 


.894934 


30 


40 


.448799 


2.2281681 


.502219 


I.99II637 


I.I 1903 


.893633 


20 


SO 


•451397 


2.2153460 


.505867 


1.9768050 


1. 12067 


.892323 


10 




• 

Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant 


Sine. 


Angle. 



SINES, COSINES, &C OF ANGLES. 



107 



Aflgk. 


Sine. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 




27*^0' 


•453990 


2.2026893 


•509525 


1. 9626105 


1. 12233 ' ,891007 


63° 0' 


10 


.456580 


2.I901947 


•513195 


1.9485772 


1. 1 2400 


.889682 


50 


20 


.459166 


2.1778595 


.516876 


1.9347020 


I.I2568 


.888350 


40 


30 


.461749 


2.1656806 


.520567 


1. 9209821 


1. 12738 


.887011 


30 


40 


.464327 


2.1536553 


.524270 


I.9074147 


1. 12910 


.885664 


20 


50 


.466901 


2.I417808 


.527984 


1.893997 1 


1. 13083 


.884309 


10 


28 


.469472 


2.1300545 


.531709 


1.8807265 


I.I3257 


.882948 


62 


10 


.472038 


2.II84737 


•535547 


1,8676003 


I- 13433 


.881578 


50 


20 


.474600 


2.1070359 


•539195 


I.8546159 


1. 13610 


.880201 


40 


30 


.477159 


2.0957385 


.542956 


I.8417409 


I.I3789 


.878817 


30 


40 


•479713 


2.0845792 


.546728 


1.8290628 


1. 13970 


.877425 


20 


50 


.482263 


2.0735556 


•550515 


1. 8164892 


I.I4152 


.876026 


10 


2g 


.4848x0 


2.0626653 


.554309 


1.8040478 


1-14335 


.874620 


61 


10 


.487352 


2.0519061 


.558118 


1. 7917362 


1.14521 


.873206 


50 


20 


.4S989O 


2.0412757 


•561939 


1.7795524 


1.14707 


.871784 


40 


30 


.492424 


2.0307720 


.565773 


1.7674940 


1. 14896 


.870356 


30 


40 


•494953 


2.0203929 


.569619 


1.7555590 


1.15085 


.868920 


20 


50 


•497479 


2.OIOI362 


.573478 


'.7437453 


1.15277 


.867476 


10 


30 


.500000 


2.0000000 


.577350 


1.7320508 


i.15470 


.866025 


60 


10 


•502517 


T.9899822 


.581235 


1.7204736 


1.T5665 


.864567 


50 


20 


.505030 


T. 98008 10 


•585134 


I.709OT16 


1.15861 


.863102 


40 


30 


•507538 


1.9702944 


.589045 


I.697663I 


1. 16059 


.861629 


30 


40 


.510043 


1.9606206 


.592970 


1.6864261 


1.16259 


.860149 


20 


50 


•512543 


1.95^0577 


.596908 


1.6752988 


1.16460 


.858662 


10 


31 


.515038 


1. 9416040 


.600861 


1.6642795 


1.16663 


.857167 


59 


10 


•517529 


1.9322578 


.604827 


1.6533663 


1.16868 


.855665 


50 


20 


.520016 


I.923OI73 


.608807 


1.6425576 


1.17075 


.854156 


40 


30 


.522499 


1.9138809* 


.612801 


1.6318517 


1.17283 


.852640 


30 


40 


•524977 


1.9048469 


.616809 


1.6212469 


1.17493 


.851117 


20 


50 


.527450 


I.8959138 


.620832 


1. 6107417 


1.17704 


.849586 


10 


32 


.529919 


1.8870799 


.624869 


1.6003345 


1.17918 


.848048 


58 


10 


.532384 


1.8783438 


.628921 


1.5900238 


1.18133 


.846503 


50 


20 


.534844 


1.8697040 


.632988 


1.5798079 


1.18350 


.844951 


40 


30 


•537300 


i.8611590 


.637079 


1.5696856 


1.18569 


.843391 


30 


40 


•539751 


1.8527073 


.641167 


1-5596552 


1.18790 


.841.825 


20 


50 


.542197 


1.8443476 


.645280 


'•5497155 


1. 19012 


.840251 


. 10 


33 


.544639 


1.8360785 


.649408 


1.5398650 


1.19236 


.838671 


57 


10 


.547076 


1.8278985 


.653531 


1. 5301025 


1. 19463 


-837083 


50 


20 


.549509 


I.8198065 


.657710 


1. 5204261 


1. 19691 


.835488 


. 40 


30 


•551937 


I.81180IO 


.661886 


I.5108352 


1.19920 


.833886 


30 


40 


.554360 


1.8038809 


.666077 


I.5013282 


1.20152 


.832277 


20 


50 


.556779 


1.7960449 


.670285 


I.4919039 


1.20386 


.830661 


10 


OxBoe. 


Secant. 


Cotangent. 


Tangent 


Cosecant. 


1 
Sine. 


Angle. 



io8 



MATHEMATICAL TABLES. 



Angle. 


Sine. 


Cosecant 


Tangent. 


Cotangent. 


Secant. 


Co^e. 


• 




34° o' 


•559193 


1.7882916 


•674509 


1. 4825610 


1.20622 


.829038 


56° 0' 




lO 


.561602 


1.7806201 


•678749 


1.4732983 


1.20859 


.827407 


50 




20 


.564007 


1.7730290 


.683007 


I.464II47 


1. 21099 


.825770 


40 




30 


.566406 


I.7655173 


.687281 


1.4550090 


1.21341 


.824126 


30 




40 


.568801 


1.7580837 


.691573 


I.445980I 


I.21584 


.822475 


20 




50 


•571191 


1.7507273 


.695881 


1.4370268 


1. 21830 


.820817 


10 




35 


•573576 


1.7434468 


.700208 


1. 4281480 


1.22077 


.819152 


55 




10 


•575957 


1. 7362413 


.704552 


I.4193427 


1.22327 


.817480 


50 




20 


•578332 


1. 7291096 


.708913 


1. 4106098 


1.22579 


.815801 


40 




30 


.580703 


1.7220508 


.713293 


I.4OI9483 


1.22833 


.814116 


30 




40 


.583069 


I.7150639 


.717691 


1-3933571 


1.23089 


.812423 


20 




50 


•5S5429 


1. 7081478 


.722108 


1.3848355 


1.23347 


.810723 


10 




36 


•587785 


I.7013016 


.726543 


1.3763810 


1.23607 


.809017 


54 




10 


.590136 


1.6945244 


.730996 


^•3679959 


1.23869 


.807304 


50 




20 


.592482 


I.687815I 


.735469 


1.3596764 


1. 24134 


.805584 


40 




30 


.594823 


1. 681 1730 


.739961 


1.3514224 


1.24400 


.803857 


30 




40 


•597159 


1.6745970 


.744472 


1.3432331 


1.24669 


.802123 


20 




50 


.599489 


1.6680864 


.749003 


1.3351075 


1.24940 


.800383 


10 




37 


.601815 


I.661640I 


.753554 


1.3270448 


1. 25214 


.798636 


53 




10 


.604136 


1-6552575 


.758125 


1.3190441 


1.25489 


.796882 


50 




20 


.606451 


1.6489376 


.762716 


1.3 1 1 1046 


1.25767 


.795121 


40 




30 


.608761 


1.6426796 


.767627 


1.3032254 


1.26047 


•793353 


30 




40 


.611067 


1.6364828 


.771959 


1.2954057 


1.26330 


.791579 


20 




50 


.613367 


1.6303462 


.776612 


1.2876447 


1. 26615 


.789798 


10 




38 


.615661 


1.6242692 


.781286 


1.2799416 


1.26902 


.788011 


52 




10 


.617951 


I.618251O 


.785981 


1.2722957 


I.2719I 


.786217 


50 




20 


.620235 


1. 6122908 


.790698 


1.2647062 


1.27483 


.784416 


40 




30 


.622515 


1.6063879 


.795436 


1.2571723 


1.27778 


.782608 


30 


40 


.624789 


1. 6005416 


.800196 


1.2496933 


1.28075 


.780794 


20 


50 


.627057 


I.594751I 


.804080 


1.2422685 


1.28374 


.778973 


10 




39 


.629320 


I.589OI57 


.809784 


1.2348972 


1.28676 


.777146 


51 




10 


•631578 


1.5833318 


.814612 


1.2275786 


1.28980 


.775312 


50 




20 


.633831 


1-5777077 


.819463 


1.2203121 


1.29287 


.773472 


40 




30 


.636078 


1.5721337 


.824336 


1. 2130970 


1.29597 


.771625 


30 




40 


.638320 


I.566612I 


.829234 


1.2059327 


1.29909 


•769771 


20 




SO 


.640557 


I.5611424 


.834155 


1.1988184 


1.30223 


.767911 


10 




40 


.642788 


1.5557238 


.839100 


1.1917536 


I.3054I 


.766044 


50 




10 


.645013 


1.5503558 


.844069 


1. 1847376 


1. 30861 


.764171 


50 




20 


.647233 


1.5450378 


.849062 


1.1777698 


1.3^83 


.762292 


40 




30 


.649448 


1.5397690 


.854081 


1. 1 708496 


I.31509 


.760406 


30 




40 


•651657 


1.5345491 


.859124 


1. 1639763 


1.31837 


.758514 


20 




50 


.653861 


1.5293773 


.864193 


1.1571495 


I.32168 


.756615 


10 






Cosine. 


Secant. 


Cotangent. 


Tangent. 


Cosecant. 


Sine. 


Angle. 





SINES, COSINES, &c. OF ANGLES. 



109 



Angle. 

1 


Sixie. 


Cosecant. 


Tangent. 


Cotangent. 


Secant. 


Cosine. 

« 




41° 0' 


.656059 


I.524253I 


.869287 


1. 1503684 


1.32501 


•754710 


49° 0' 


10 


.658252 


I.519I759 


.874407 


1. 1436326 


1.32838 


.752798 


50 


20 


.660439 


I.514I452 


.879553 


I.I369414 


I.33177 


.750880 


40 


30 


.662620 


I.5091605 


.884725 


1. 1302944 


I.33519 


.748956 


30 


40 


.664796 


I.50422II 


.889924 


1. 1236909 


1.33864 


.747025 


20 


50 


.666966 


1.4993267 


.895151 


I.II71305 


1.34212 


.745088 


10 


142 


.669131 


1.4944765 


.900404 


I.II06125 


1.34563 


.743145 


48 


ID 


.671289 


1.4896703 


.905685 


I.IO41365 


I.34917 


.741195 


50 


20 


.673443 


1.4849073 


.910994 


1.0977020 


1.35274 


.739239 


40 


30 


.675590 


1. 4801872 


.916331 


I.O913085 


1.35634 


.737277 


30 


40 


.677732 


1.4755095 


.921697 


1.0849554 


1-35997 


.735309 


20 


50 


.679868 


1.4708736 


.927091 


1.0786423 


1.36363 


.733335 


10 


43 ex 


.681998 


1.4662792 


.932515 


1.0723687 


1.36733 


.731354 


47 


10 


.684123 


I.4617257 


.937968 


I.066134I 


1-37105 


.729367 


50 


20 


.686242 


I.4572127 


.943451 


I.0599381 


1.37481 


.727374 


40 


30 


.688355 


1.4527397 


.948965 


I.O53780I 


1.37860 


.725374 


30 


40 


.690462 


1.4483063 


.954508 


1.0476598 


1.38242 


.723369 


20 


50 


.692563 


1.4439 1 20 


.960083 


I.O415767 


1.38628 


.721357 


10 


44 


.694658 


1.4395565 


.965689 


1.0355303 


1.39016 


.719340 


46 


10 


.696748 


1.4352393 


.971326 


1.0295203 


1.39409 


.717316 


50 


20 


.698832 


1.4309602 


.976996 


I.O235461 


1.39804 


.715286 


40 


30 


. 700909 


1. 4267182 


.982697 


i.oi 76074 


1.40203 


.713251 


30 


40 


.702981 


I.4225134 


.988432 


I.OII7088 


1.40606 


.711209 


20 


i 50 

1 


.705047 


1.4183454 


.994199 


1.0058348 


1.41012 


.709161 


10 


45 


.707107 


I.4142136 


1. 000000 


1. 0000000 


1.41421 


.707107 


45 


Cosine. 


SecanL 


Cotangent. 


Tangent 


Cosecant. 


Sine. 


Angle. 



' 



no 



MATHEMATICAL TABLES. 



TABLE No. IX.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS OF ANGLES FROM o^ TO 90^ 

Advancing by ic/, or one-sixth of a degree. 



Angle. 


Sine. 


Tangent 


Cotangent. 


. Cosine. 




0° 


0.000000 


0.000000 


Infinite. 


10.000000 


9o« 


lo' 


7.463726 


7.463727 


".536273 


9.999998 


50' 


20 


7.764754 


7.764761 


12.235239 


9.999993 


40 


30 


7.940842 


7.940858 


12.059142 


9.999983 


30 


40 


8.065776 


8.065806 


11.934194 


9.999971 


20 


50 


8.162681 


8.162727 


11.837273 


9.9999S4 


10 


I 


8.241855 


8.24192I 


11.758079 


9.999934 


89 


10 


8.308794 


8.308884 


11.691116 


9.999910 


50 


20 


8.366777 


8.366895 


11.633105 


9.999882 


40 


30 


8.417919 


8.418068 


11.58x932 


9.999851 


30 


40 


8.463665 


8.463849 


11.536151 


9.999816 


20 


50 


8.505045 


8.505267 


11.494733 


9.999778 


10 


2 


8.542819 


8.543084 


11.456916 


9.999735 


88 


10 


8.577566 


8.577877 


11.422123 


9.999689 


50 


20 


8.609734 


8.610094 


11.389906 


9.999640 


40 


30 


8.639680 


8.640093 


11.359907 


9.999586 


30 


40 


8.667689 


8.668160 


11.331840 


9.999529 


20 


50 


8.693998 


8.694529 


11.305471 


9.999469 


10 


3 


8.718800 


8.719396 


11.280604 


9.999404 


87 


10 


8.742259 


8.742922 


11.257078 


9.999336 


50 


20 


8. 7645 II 


8.765246 


11.234754 


9.999265 


40 


30 


8.785675 


8.786486 


11.213514 


9.999189 


30 


40 


8.805852 


8.806742 


11.193258 


9.999110 


20 


SO 


8.825130 


8.826103 


11.173897 


9.999027 


10 


4 


8.843585 


8.844644 


11.155356 


9.998941 


86 


10 


8.861283 


8.862433 


II. 137567 


9.998851 


50 


20 


8.878285 


8.879529 


1 1. 1 2047 1 


9.998757 


40 


30 


8.894643 


8.895984 


11.104016 


9.998659 


30 


40 


8.910404 


8.9I1846 


11.088154 


9.998558 


20 


SO 


8.925609 


8.927156 


11.072844 


9.998453 


10 


5 


8.940296 


8.941952 


11.058048 


9.998344 


85 


10 


8.954499 


8.956267 


11.043733 


9.998232 


50 


20 


8.968249 


8.970133 


11.029867 


9.9981 16 


40 


30 


8.981573 


8.983577 


II. 016423 


9.997996 


30 


40 


8.994497 


8.996624 


11.003376 


9.997872 


20 


so 


9.007044 


9.009298 


10.990702 


9.997745 


10 




Cosine. 


Cotangent. 


Tangent 


Sine. 


Angle. 



LOGARITHMIC SINES, TANGENTS, &C. 



Ill 



Aogie. 


1 

1 Sine. 

1 


Tangent 


Cotangent. 


Cosine. 




6^ 


9.019235 


9.021620 


10.978380 


9.997614 


84° 


lo' 


9,031089 


9.O33JS09 


10.966391 


9.997480 


50' 


20 


9.042625 


9.045284 


10.954716 


9.997341 


40 


30 


9-053859 


9.056659 


10.943341 


9.997199 


30 


40 


9.064806 


9.067752 


10.932248 


9.997053 


20 


50 


9.075480 


9.078576 


10.921424 


9.996904 


10 


7 


9.085894 


9.089144 


10.910856 


9.996751 


83 


10 


9.096062 


9.099468 


10.900532 


9.996594 


50 


20 


9.105992 


9.109559 


10.890441 


9.996433 


40 


30 


9. 1 15698 


9.II9429 


10.880571 


9.996269 


30 


40 


1 9-125187 


9.129087 


10.870913 


9.996100 


20 


50 


9.134470 


9138542 


10.861458 


9.995928 


10 


8 


9-I43SS5 


9.147803 


10.852197 


9.995753 


82 


10 


9-152451 


9.156877 


10.843123 


9.995573 


50 


20 


9.161164 


9-165774 


10.834226 


9.995390 


40 


30 


9.169702 


9.174499 


10.825501 


9.995203 


30 


40 


9.178072 


9-183059 


IO.81694I 


9.995013 


20 


50 


9.186280 


9.I91462 


10.808538 


9.994818 


10 


9 


9-194332 


9199713 


10.800287 


9.994620 


81 


10 


9.202234 


9.207817 


10.792183 


9.994418 


50 


20 


9.209992 


9.215780 


10.784220 


9.994212 


40 


30 


9.217609 


9.223607 


10.776393 


9.994003 


30 


40 


9.225092 


9.231302 


10.768698 


9.993789 


20. 


50 


9.232444 


9.238872 


IO.761128 


9-993572 


10 


10 


9.239670 


9.246319 


10.753681 


9.993351 


80 


10 


9.246775 


9.253648 


10.746352 


9.993127 


50 


20 


' 9-253761 


9.260863 


10.739137 


9.992898 


40 


3<^ 


9.260633 


9.267967 


10.732033 


9.992666 


30 


40 


9-2673,95 


9.274964 


10.725036 


9.992430 


20 


50 


9.274049 


9.281858 


IO.718142 


9.992190 


10 


II 


9.280599 


9.288652 


10.71 1348 


9.991947 


79 


10 


9.287048 


9.295349 


10.704651 


9.991699 


50 


20 


9-293399 


9-301951 


10.698049 


9.991448 


40 


30 


9-299655 


9.308463 


10.691537 


9.991 193 


30 


40 


9-305819 


9.314885 


10.6851 15 


9.990934 


20 


50 


9-31 1893 


9.321222 


10.678778 


9.990671 


10 


12 


9.317879 


9.327475 


10.672525 


9.990404 


78 


10 


9.323780 


9-333646 


10.666354 


9.990134 


50 


20 


9-329599 


9.339739 


10.660261 


9.989860 


40 


30 


9-335337 


9.34575s 


10.654245 


9.989582 


30 


40 


9-340996 


9.351697 


10.648303 


9.989300 


20 


50 


9-346779 


9.357566 


10.642434 


9.989014 


10 




Cosine. 


Cotangent. 


Tangent 


Sine. 


Angle. 






112 



MATHEMATICAL TABLES. 



Angle. 


Sine. 


Tangent. 


Cotangent. 


Cosine. 


1 


13° 


9.352088 


9.363364 


10.636636 


9.988724 


If 


10' 


9-357524 


9-369094 


10.630906 


9.988430 


so' 


20 


9.362889 


9.374756 


10.625244 


9-988133 


40 


30 


9.368185 


9380354 


10.619646 


9.987832 


30 


40 


9-373414 


9.385888 


IO.614II2 


9.987526 


20 


50 


9-378577 


9.391360 


10.608640 


9.987217 


10 


14 


9-383675 


9.396771 


10.603229 


9.986904 


76 


10 


9.388711 


9.402124 


10.597876 


9.986587 


50 


20 


9-393685 


9.407419 


10.592581 


9.986266 


40 


30 


9.398600 


9.412658 


10.587342 


9.985942 


30 


40 


9-403455 


9.417842 


10.582158 


9.985613 


20 


50 


9.408254 


9.422974 


10.577026 


9.985280 


10 


15 


9.412996 


9.428052 


10.571948 


9-984944 


75 


10 


9.417684 


9.433080 


10.566920 


9.984603 


50 


20 


9.422318 


9438059 


IO.56194I 


9-984259 


40 


30 


9.426899 


9.442988 


10.557012 


9.9839 1 1 


30 


40 


9.431429 


9.447870 


10.552130 


9-983558 


20 


50 


9-435908 


9.452706 


10.547294 


9.983202 


10 


16 


9440338 


9-457496 


10.542504 


9.982842 


74 


10 


9.444720 


9.462242 


10.537758 


9.982477 


50 


20 


9.449054 


9.466945 


10.533055 


9.982109 


40 


30 


9-453342 


9.471605 


10.528395 


9-981737 


30 


40 


9-457584 


9.476223 


10.523777 


9.981361 


20 


50 


9.461782 


9.480801 


IO.519199 


9.980981 


10 


17 


9-465935 


9.485339 


1 0.5 1 466 1 


9.980596 


73 


10 


9.470046 


9-489838 


IO.51O162 


9.980208 


50 


20 


9-4741 15 


9-494299 


10.505701 


9.979816 


40 


30 


9.478142 


9.498722 


10.501278 


9-979420 


30 


40 


9.482128 


9.503109 


10.496891 


9.979019 


20 


50 


9.48607s 


9.507460 


10.492540 


9.978615 


10 


18 


9.489982 


9.5II776 


10.488224 


9.978206 


72 


10 


9.493851 


9-516057 


10.483943 


9-977794 


50 


20 


9.497682 


9.520305 


10.479695 


9-977377 


40 


30 


9.501476 


9.524520 


10.475480 


9-976957 


30 


40 


9.505234 


9.528702 


10.471298 


9-976532 


20 


50 


9-508956 


9.532853 


10.467147 


9.976103 


10 


19 


9.512642 


9.536972 


10.463028 


9.975670 


71 


10 


9.516294 


9.541061 


10.458939 


9.975233 


50 


20 


9.519911 


9.545II9 


10.454881 


9.974792 


40 


30 


9.523495 


9.549149 


10.450851 


9.974347 


30 


40 


9.527046 


9.553149 


10.446851 


9.973897 


20 


50 


9.530565 


9.55712I 


10.442879 


9-973444 


10 




Cosine. 


Cotangent 


Tangent. 


Sine. 


Angle. 







LOGARITHMIC SINES, TANGENTS, &C 


"3 




Ai«le. 


Sine. 


Tangent. 


Cotangent. 


Cosme. 




20^ 


9-534052 


9.561066 


10.438934 


9.972986 


700 




10' 


9537507 


9.564983 


10.435017 


9.972524 


50' 




20 


9540931 


9.568873 


IO.43II27 


9.972058 


40 




30 


9-544325 


9.572738 


10.427262 


9.971588 


30 




40 


9.547689 


9.576576 


10.423424 


9.971113 


20 




50 


9-551024 


9.580389 


IO.41961I 


9.970635 


10 




21 


9-554329 


9.584177 


10.415823 


9.970152 


69 




10 


9.557606 


9.587941 


10.412059 


9.969665 


50 




20 


9.56085s 


9.591681 


10.408319 


9.969173 


40 




30 


9.564075 


9.595398 


10.404602 


9.968678 


30 




40 


9.567269 


9.599091 


10.400909 


9.968178 


20 




50 


9.570435 


9.602761 


10.397239 


9.967674 


10 




22 


9-573575 


9.606410 


10.393590 


9.967166 


68 




10 


9.576689 


9.610036 


10.389964 


9.966653 


50 




20 


9.579777 


9.61364I 


10.386359 


9.966136 


40 




30 


9.582840 


9.617224 


10.382776 


9-965615 


30 




40 


9.585877 


9.620787 


10.379213 


9.965090 


20 




50 


9.588890 


9.624330 


10.375670 


9.964560 


10 




23 


9.591878 


9.627852 


10.372148 


9.964026 


67 




10 


9.594842 


9.631355 


10.368645 


9.963488 


50 




20 


9.597783 


9.634838 


10.365162 


9.962945 


40 




30 


9.600700 


9.638302 


10.361698 


9.962398 


30 




40 


9.603594 


9.641747 


10.358253 


9.961846 


20 




50 


9.606465 


9.645174 


10.354826 


9.961290 


10 




M 


9.609313 


9.648583 


IO.351417 


9.960730 


66 




10 


9.6 1 2 140 


9.651974 


10.348026 


9.960165 


SO 




20 


9.614944 


9.655348 


10.344652 


9-959596 


40 




30 


9.617727 


9.658704 


10.341296 


9-959023 


30 




40 


9.620488 


9.662043 


10.337957 


9.958445 


20 




50 


9.623229 


9.665366 


10.334634 


9.957863 


10 




25 


9.625948 


9.668673 


10.331328 


9.957276 


65 




10 


9.628647 


9.671963 


10.328037 


9.956684 


50 




20 


9.631326 


9.675237 


10.324763 


9.956089 


40 




30 


9.633984 


9.678496 


10.321504 


9-955488 


30 




40 


9.636623 


9.681740 


10.318260 


9-954883 


20 




50 


9.639242 


9.684968 


10.315032 


9-954274 


10 




26 


9.641842 


9.688182 


IO.3I1818 


9-953660 


64 




10 


9.644423 


9-691381 


10.308619 


9-953042 


SO 




20 


9.646984 


9.694566 


10.305434 


9.952419 


40 




30 


9.649527 


9.697736 


10.302264 


9-951791 


30 




40 


9.652052 


9.700893 


10.299107 


9-95"S9 


20 




50 


9.654558 


9.704036 


10.295964 


9.950522 


10 






Cosine. 


Cotangent. 


Tangent 


Sine. 


Angle. 



8 



"4 



MATHEMATICAL TABLES. 



Angle. 


Sine. 


Tangent. 


Cotangent. 


Cosine. 




27° 


9.657047 


9.707166 


10.292834 


9.949881 


63^ 


10' 


9-659517 


9.710282 


10.289718 


9-949235 


50' 


20 


9.661970 


9.713386 


10.286614 


9.948584 


40 


30 


9.664406 


9.716477 


10.283523* 


9.947929 


30 


40 


9.666824 


9-719555 


10.280445 


9.947269 


20 


50 


9.669225 


9.722621 


10.277379 


9.946604 


10 


28 


9.671609 


9.725674 


10.274326 


9-945935 


62 


10 


9-673977 


9.728716 


10.271284 


9.945261 


50 


20 


9.676328 


9-731746 


10.268254 


9.944582 


40 


30 


9.678663 


9-734764 


10.265236 


9-943899 


30 


40 


9.680982 


9-737771 


10.262229 


9.943210 


20 


50 


9.683284 


9.740767 


10.259233 


9-942517 


10 


29 


9-685571 


9-743752 


10.256248 


9.941819 


61 


10 


9.687843 


9.746726 


10.253274 


9.941117 


50 


20 


9.690098 


9.749689 


IO.2503II 


9.940409 


40 


30 


9.692339 


9.752642 


10.247358 


9.939697 


30 


40 


9.694564 


9-755585 


10.244415 


9.938980 


20 


50 


9.696775 


9.758517 


10.241483 


9-938258 


10 


30 


9.698970 


9.761439 


10.238561 


9-937531' 


60 


10 


9.70II51 


9.764352 


10.235648 


9.936799 


50 


20 


9-703317 


9-767255 


10.232745 


.9.936062 


40 


30 


9.705469 


9.770148 


10.229852 


9-935320 


30 


40 


9.707606 


9.773033 


10.226967 


9-934574 


20 


SO 


9.709730 


9-775908 


10.224092 


9.933822 


10 


31 


9.711839 


9-778774 


10.221226 


9.933066 


59 


10 


9-713935 


9.781631 


10.218369 


9.932304 


50 


20 


9.716017 


9-784479 


IO.21552I 


9931537 


40 


30 


9-718085 


9-787319 


IO.212681 


9.930766 


30 


40 


9.720140 


9.790151 


10.209849 


9.929989 


20 


50 


9.722181 


9-792974 


10.207026 


9.929207 


10 


32 


9.724210 


9.795789 


10.2042 1 1 


9.928420 


58 


10 


9.726225 


9.798596 


10.201404 


9.927629 


50 


20 


9.728227 


9.801396 


10.198604 


9.926831 


40 


30 


9.730217 


9.804187 


IO.I95813 


9.926029 


30 


40 


9732193 


9.806971 


10.193029 


9.925222 


20 


50 


9-734157 


9.809748 


10.190252 


9.924409 


10 


33 


9.736109 


9.812517 


10.187483 


9-923591 


57 


10 


9.738048 


9.815280 


10.184720 


9.922768 


50 


20 


9-739975 


9.818035 


IO.181965 


9.921940 


40 


30 


9.741889 


9.820783 


IO.I79217 


9.921107 


30 


40 


9-743792 


9-823524 


10.176476 


9.92026^ 


20 


50 


9.745683 


9.826259 


IO.I7374I 


9.919424 


10 




Cosine. 


Cotangent. 


Tangent. 


Sine. 


Angle. 



LOGARITHMIC SINES, TANGENTS, &C 



115 



Angle. 


Sine. 


Tangent 


Cotangent 


Cosine. 




' 34 


9.747562 


9.828987 


IO.1710I3 


9.918574 


56° , 


10' 


9.749429 


9.831709 


. IO.16829I 


9.917719 


50' 


20 


9.751284 


9-834425 


10.165575 


9.916859 


40 


30 


9753128 


9-837134 


10.162866 


9-915994 


30 


40 


9.754960 


9.839838 


IO.160162 


9-915123 


20 


50 


9.756782 


9-842535 


10.157465 


9.914246 


10 


35 


9-758591 


9.845227 


10.154773 


9.913365 


55 


10 


9.760390 


9.847913 


10.152087 


9.912477 


50 


20 


9.762177 


9-850593 


10.149407 


9.91 1584 


40 


30 


9-763954 


9.853268 


10.146732 


9.910686 


30 


40 


9.765720 


9-855938 


10.144062 


9.909782 


20 


50 


9-767475 


9.858602 


IO.I41398 


9.908873 


10 


36 


9.769219 


9.861261 


10.138739 


9.907958 


54 


10 


9.770952 


9-863915 


10.136085 


9.907037 


50 


20 


9.77267s 


9.866564 


10.133436 


9.9061 11 


40 


30 


9-774388 


9.869209 


IO.I3079X 


9.905179 


30 


40 


9.776090 


9.871849 


IO.I28151 


9.904241 


20 


50 

1 


9-777781 


9.874484 


IO.I25516 


9.903298 


10 


1 

37 


9-779463 


9.877114 


10.122886 


9.902349 


53 


10 


1 9-781134 


9.879741 


10.120259 


9.901394 


50 


20 


1 9-782796 


9.882363 


IO.II7637 


9.900433 


40 


30 


9.784447 


9.884980 


10. 1 1 5020 


9.899467 


30 


40 


9.786089 


9.887594 


10. 1 12406 


9-898494 


20 


50 


9.787720 


9.890204 


10.109796 


9.897516 


10 


38 


9.789342 


9.892810 


IO.IO719O . 


9.896532 


52 


10 


9-790954 


9.895412 


10.104588 


9.895542 


50 


20 


9792557 


9.898010 


IO.IOI99O 


9.894546 


40 


30 


9-794150 


9.900605 


10.099395 


9.893344 


30 


40 


9-795733 


9.903197 


10.096803 


9.892536 


20 


50 


9.797307 


9-905785 


10.094215 


9.891523 


10 


39 


9-798872 


9.908369 


IO.O9163I 


9.890503 


51 


10 


9.800427 


9.910951 


10.089049 


9.889477 


50 


20 


9.801973 


9-913529 


10.086471 


9.888444 


40 


1 3<^ 


9.803511 


9.916104 


10.083896 


9.887406 


30 


1 ^^ 
40 


9.805039 


9.918677 


10.081323 


9.886362 


20 


50 


9-806557 


9.921247 


10.078753 


9.885311 


10 


40 


9.808067 


9.923814 


10.076186 


9.884254 


50 


10 


9.809569 


9.926378 


10.073622 


9.883191 


50 


20 


9.811061 


9.928940 


10.071060 


9.88212I 


40 


30 


9.812544 


9-931499 


10.068501 


9.881046 


30 


40 


9.814019 


9-934056 


10.065944 


9.879963 


20 


50 


9-815485 


9.93661 1 


10.063389 


9.878875 


10 




Coune. 


Cotangent. 


T.ingcnt. 


Sine. 


Angle. 



.:j,-«rrf'^f-" 



ri6 



MATHEMATICAL TABLES. 



Angle. 


Sine. 


Tangent 


Cotangent. 


Cosine. 


1 


41° 


9.816943 


9-939163 


10.060837 


9.877780 


49° 


10' 


9.818392 


9-941713 


10.058287 


9.876678 


50' 


20 


9.819832 


9.944262 


10.055738 


9-875571 


40 


30 


9.821265 


9.946808 


10.053192 


9.874456 


30 


40 


9.822688 


9-949353 


10.050647 


9.873335 


20 


50 


9.824104 


9.951896 


10.048104 


9.872208 


10 


42 


9-825511 


9-954437 


10.045563 


9.871073 


48 


10 


9.826910 


9-956977 


10.043023 


9.869933 


50 


20 


9.828301 


9-959516 


10.040484 


9.868785 


40 


30 


9.829683 


9.962052 


10.037948 


9.867631 


30 


40 


9.831058 


9.964588 


10.035412 


9.866470 


20 


50 


9-832425 


9.967123 


10.032877 


9.865302 


10 


43 


9-833783 


9.969656 


10.030344 


9.864127 


47 


10 


9-835134 


9,972188 


10.027812 


9.862946 


50 


20 


9.836477 


9.974720 


10.025280 


9-861758 


40 


30 


9.837812 


9.977250 


10.022750 


9.860562 


30 


40 


9.839140 


9.979780 


10.020220 


9.859360 


20 


50 


9.840459 


9.982309 


10.01 7691 


9.85815I 


10 


44 


9.841771 


9-984837 


IO.OI5163 


9-856934 


46 


10 


9.843076 


9-987365 


10.012635 


9.855711 


SO 


20 


9.844372 


9.989893 


IO.OIOIO7 


9.854480 


40 


30 


9.845662 


9.992420 


10.007580 


9-853242 


30 


40 


9.846944 


9.994947 


10.005053 


9-851997 


20 


SO 


9.848218 


9-997473 


10.002527 


9850745 


10 


45 


9.849485 


10.000000 


10.000000 


9.849485 


45 




Cosine. 


Cotangent. 


Tangent. 


Sine. 


Angle. 



RHUMBS, OR POINTS OF THE COMPASS. 



117 



TABLE No. X.— RHUMBS, OR POINTS OF THE COMPASS. 



Points. 



H 
>i 
Va- 

I 

2 

3 

sH 
4 

4H 






6^ 



8 



Angles. 



2^48' 45' 

5 37 30 

8 26 15 

II 15 o 

U 3 45 

16 52 30 

19 41 15 

22 30 o 

25 18 45 

28 7 30 

30 56 15 

33 45 o 

36 33 45 

39 22 30 

42 n 15 

45 o o 

47 48 45 

50 37 30 

53 26 15 

56 15 o 

59 3 45 

61 52 30 

64 41 15 

67 30 o 

70 18 45 

73 7 30 

75 56 15 

78 45 o 

81 ZZ 45 

84 22 30 

87 II 15 
90 o o 



NORTH. 



N ^ E 

N 5^ 



- /* E 
N ^ E 

N by E 



Nby E ;^ 

N by E J^ E 
N by E ^ E 

NNE 

NNE ^ E 

NNE ^ E 

NNE ^ E 

NE by N 
NE^N 

'' N 
N 



NE>52 



NE 
NE J^ E 
NE J^ E 
NE J^ E 

NE by E 
ENE ^ N 
ENE y2 N 
ENE ]/{ N 

ENE 

ENE ^ E 

ENE ^ E 

ENE y^ E 

Eby N 
E% N 
E li N 



E^N 
EAST. 



NORTH. 



SOUTH. 



N ^ W 
N ^ W 
N J^ W 

N by W 
N by w ^ w 
N by w J^ w 
N by w ^ w 

NNW 
NNW j5^ W 
NNW j5 W 

NNW y^ W 

NW by N 
NW ^ N 
NW i^ N 
NW^ N 

NW. 
NW l^ W 
NW J^ W 
NW J^ W 

NW by w 

WNW y^ N 
WNW ^ N 
WNW J^ N 

WNW 
WNW ^ W 



WNW J^ W 

WNW y^ W 

. w by N 

W^ N 
w J4 N 
W 5j( N 

WEST. 



S )4 E 
S >^ E 
S^E 

s by E 



E 



sby E 

s by E ^ E 

s by £ )^ E 

SSE 
SSE ^ E 
SSE Yi E 
SSE J^ E 

SE by s 

SE ^ S 
SE ^ S 
SE ^ S 

SE 
SE l^ E 
SE ^ E 
SE ^ E 

SE by E 
ESE ^ S 

ESE y2 S 

ESE^ S 

ESE 
ESE % E 
ESE J4 E 
ESE y^ E 

E by S 
E ^ S 
E^ S 
E j^ S 

EAST. 



SOUTH. 



S 

s 



s^w 
s>^ w 

s^w 

s by w 
sby w ^ w 
by w J^ w 
by w ^ w 

ssw 
ssw ^ w 
ssw y^ w 

ssw y^ w 

sw by s 
sw^ s 
sw ^ s 
sw i^ s 

sw 
sw^ w 
sw J4 w 
sw ^ w 

swby w 
wsw y^ s 
wsw J4 s 
wsw^ s 

wsw 
wsw 5^ w 
wsw J^ w 

wsw y^ w 

w by s 
w^ s 
w J4s 
w j^ s 

WEST. 



MATHEMATICAL TABLES. 



TABLE No. XL— RECIPROCALS OF NUMBERS 



RECIPROCALS OF NUMBERS. 



119 



No. 

57 
58 
59 

[60 

61 
62 

63 
64 

66 

67 
68 

69 

70 
71 
72 

73 
74 

75 
76 

77 
73 

79 

80 

81 
82 

84 

85 
86 

87 
88 
89 

90 

91 
92 

93 
94 

95 
96 

97 

98 

99 

200 

201 I 



Reciprocal. 



006369 
006329 
006289 

006250 
0062 I I 
006173 
006135 
006098 
006061 
006024 
005988 
005952 
005917 

005882 
005848 
005814 
005780 

005747 
005714 
005682 

005650 
005618 
005587 

005556 
005225 

005495 

005464 

005435 
005405 

005376 

005348 

005319 

005291 

005263 
005236 
005208 
OO5181 

005155 
005128 

005102 

005076 

005051 

005025 

005000 
004975 



No. 



202 
203 
204 
205 
206 
207 
208 
209 

210 

211 
212 
213 
214 

215 
216 
217 
218 
219 

220 

221 
222 
223 
224 
225 
226 
227 
228 
229 

230 

231 
232. 

233 
234 

235 
236 

237 
238 

239 

240 

241 
242 

243 

244 

245 
246 



ReciprocaL 



004950 
004926 
004902 
004878 
004854 
00483 I 
004808 
004785 

004762 

004739 
0047 I 7 

004695 

004673 

00465 1 

004630 

004608 

004587 

004566 

004545 
004525 

004505 

004484 

004464 

004444 

004425 

004405 

004386 

004367 

004348 
004329 
004310 
004292 
0042 74 
004255 
004237 
004219 
004202 
004184 

004167 
004149 
004132 
004115 
004098 
004082 
004065 



No. 



247 
248 

249 

250 

251 
252 

253 
254 

255 
256 

257 
258 

259 

260 
261 
262 
263 
264 
265 
266 
267 
268 
269 

270 
271 
272 

273 
274 

275 
276 

277 

278 

279 

280 

281 
282 
283 
284 
285 
286 
287 
288 
289 

290 
291 



Reciprocal. 



I 



004049 
004032 
004016 

004000 
003984 
003968 

003953 
003937 
003922 
003906 
003891 
003876 
003861 

003846 

003831 

003817 

003802 

003788 

003774 

003759 
003745 

003731 

003717 

003704 
003690 
003676 
003663 
003650 
003636 
003623 
003610 

003597 
003584 

003571 

003559 
003546 

003534 
003522 

003509 

003497 

003484 

003472 

003460 

003448 
003436 



No. 



292 

293 
294 

295 
296 

297 
298 

299 

300 

301 

302 

303 

304 

305 
306 

307 
308 

309 

310 

311 
312 

313 
314 

315 
316 

317 
318 

319 

320 

321 
322 

323 
324 

325 
326 

327 
328 

329 
330 

332 

333 
334 
335 
336 



Reciprocal. 



.003425 
.003413 
.003401 
.003390 
.003378 
.003367 
.003356 
.003344 

.003333 
.003322 
.003311 
.003301 
.003289 
.003279 
.003268 
.003257 
.003247 
.003236 

.003226 
.003215 
.003205 
.003195 
.003185 
.003175 
.003165 
.003155 
.003145 
.003135 

.003125 
.003115 
.003106 
.003096 
.003086 
.003077 
.003067 
.003058 
.003049 
.003040 

.003030 
.003021 
.003012 
.003003 
.002994 
.002985 
.002976 



I 



1 20 



MATHEMATICAL TABLES. 



No. 



337 
338 
339 



380 

381 



Reciprocal. 



.002967 
.002959 
.002950 



340 


.002941 


341 


.002933 


342 


.002924 


343 


.002915 


344 


.002907 


345 


.002899 


346 


.002890 


347 


.002882 


348 


.002874 


349 


.002865 


350 


.002857 


35J 


.002849 


352 


.002841 


353 


.002833 


354 


.002825 


355 


.002817 


356 


.002809 


357 


.002801 


358 


.002793 


359 


.002786 


360 


.002778 


361 


.002770 


362 


.002762 


363 


.002755 


364 


.002747 


365 


.002740 


366 


.002732 


367 


.002725 


368 


.002717 


369 


.002710 


370 


.002703 


371 


.002695 


372 


.002688 


373 


.002681 


374 


.002674 


375 


.002667 


376 


.002660 


377 


.002653 


378 


.002646 


379 


.002639 



.002632 
.00^625 



No. 



382 

383 

384 

385 
386 

387 
388 

389 

390 

391 
392 

393 
394 

395 
396 
397 
398 
399 

400 

401 
402 

403 

404 

405 
406 

407 

408 

409 

410 

411 
412 

413 
414 

415 
416 

417 
418 

419 

420 

421 
422 

423 

424 

425 
426 



Redprocal. 



.002618 
.002611 
.002604 
.002597 
.002591 
.002584 
.002577 
.002571 

.002564 
.002558 
.002551 
.002545 
.002538 
.002532 
.002525 
.002519 
.002513 
.002506 

.002500 
.002494 
.002488 
.002481 
.002475 
.002469 
.002463 
.002457 
.002451 
.002445 

.002439 
.002433 
.002427 
.002421 
.002415 
.002410 
.002407 
.002398 
.002392 
.002387 

.002381 
.002375 
.002370 
.002364 
.002358 
.002353 
.002347 



No. 



427 
428 

429 

430 

431 
432 

433 

434 

435 
436 

437 
438 

439 

440 
441 
442 
443 
444 

445 
446 

447 
448 

449 

450 
451 

452 
453 
454 
455 
456 
457 
458 

459 

460 

461 
462 

463 
464 

465 
466 

467 

468 

469 

470 

471 



Reciprocal. 



.002342 
.002336 
.002331 

,002326 
.002320 
.002315 
.002309 
.002304 
.002299 
.002294 
.002288 
.002283 
.002278 

.002273 
.002268 
.002262 
.002257 
.002252 
.002247 
.002242 
.002237 
.002232 
.002227 

.002222 
.002217 
.002212 
.002208 
.002203 
.002198 
.002193 
.002188 
.002183 
.002179 

.002174 
.002169 
.002165 
.002160 
.002155 
.002151 
.002146 
.002141 
,002137 
.002132 

.002128 
.002123 



No. 



472 

473 

474 

475 
476 

477 
478 

479 

480 

481 
482 

483 

484 

485 
486 

487 
488 

489 

490 

491 

492 

493 

494 

495 
496 

497 

498 

499 

500 

501 
502 

503 
504 

505 
506 

507 
508 

509 

510 

511 
512 

513 
514 

5x5 
516 



Reciprocal. 



.002119 
.002114 
.002110 
.002105 
.002101 
.002096 
.002092 
.002088 

.002083 
.002079 
.002075 
.002070 
.002066 
.002062 
.002058 
.002053 
.002049 
.002045 

.002041 
.002037 
.002033 
.002028 
.002024 
.002020 
.002016 
.002012 
.002008 
.002004 

.002000 
.001996 
.001992 
.001988 
.001984 
.001980 
.001976 
.001972 
.001969 
.001965 

.001961 
.001957 
.001953 
.001949 
.001946 
.001942 
.001938 



RECIPROCALS OF NUMBERS. 



121 



Ko. 



518 

520 

521 
522 

526 

528 
529 

532 

533 
534 

535 
536 

537 
538 
539 

540 
541 
542 

543 
544 

545 
546 

547 

548 

549 

550 

551 
552 

553 
554 

555 
556 

557 
558 
559 

560 

561 



RcaprocaL 



.001934 
.001931 
.001927 

.001923 
.001919 
.001916 
.001912 
.001908 
.001905 
.001901 
.001898 
.001894 
.001890 

.001887 
.001883 
.001880 
.001876 
.001873 
.001869 
.001866 
.001862 
.001859 
.001855 

.001852 
.001848 
.001845 
.001842 
.001838 
.001835 
.001832 
.001828 
.001825 
.001821 

.001818 
.001815 
.001812 
.001808 
.001805 
.001802 
.001799 
.001795 
.001792 
.001789 

.001786 
.001783 



No. 



562 

563 
564 

565 
566 

567 
568 

569 

571 

572 

573 
574 

575 
576 

577 
578 

579 

580 

581 
582 

583 
584 

585 
586 

587 
588 

589 

590 

591 
592 

593 
594 
595 
596 
597 
598 
599 

600 

601 
602 
603 
604 
605 
606 



RedprocaL 



.00 
.00 
.00 
.00 
.00 
.00 
.00 
.00 

.00 
.00 
.00 
.00 
.00 
.00 
.00 
.00 
.00 
.00 

.00 
.00 
.00 
.00 
.00 
.00 
.00 
.00 
.00 
.00 

.00 
.00 
.00 
.00 
.00 
.00 
.00 
,00 
.00 
.00 

.00 
.00 
.00 
.00 
.00 
.00 
.00 



79 
76 

73 
70 

67 

64 
61 

57 

54 

51 
48 

45 
42 

39 
36 

33 
30 
27 

24 
21 

18 

15 
12 

09 

06 

04 

01 

698 

695 
692 

689 

686 

684 

681 

678 

675 
672 

669 

667 
664 
661 
658 
656 

653 
650 



No. 



607 
608 
609 

610 
611 
612 
613 
614 

615 
616 

617 

618 

619 

620 
621 
622 
623 
624 
625 
626 
627 
628 
629 

630 
631 
632 

633 
634 

635 
636 

637 
638 

639 

640 

641 
642 

643 
644 

645 
646 

647 

648 

649 

650 
651 



Reciprocal. 



.001647 
.001645 
.001642 

.001639 
.001637 
.001634 
.001631 
.001629 
.001626 
.001623 
.001621 
.001618 
.001616 

.001613 
.001610 
.001608 
.001605 
.001603 
.001600 
.001597 

.001595 
.001592 

.001590 

.001587 
.001585 
.001582 
.001580 
.001577 

•001575 
.001572 

.001570 

.001567 

.001565 

.001563 
.001560 
.001558 

.001555 
•001553 
.001550 

.001548 
.001546 
.001543 
.001541 

.001538 
.001536 



No. 



Reciprocal. 



652 

653 
654 

655 
656 

657 
658 

659 

660 
661 
662 
663 
664 
665 

666 
667 
668 
669 

670 

671 
672 

673 
674 

675 
676 

677 

678 

679 

680 

681 
682 
683 
684 
685 
686 
687 
688 
689 

690 

691 
692 

693 
694 

695 
696 



.001534 
.001531 
.001529 
.001527 
.001524 
.001522 
.001520 
.001517 

.001515 
.001513 
.001511 
.001508 
.001506 
.001504 
.001502 
.001499 
.001497 
.001495 

.001493 
.001490 
.001488 
.001486 
.001484 
.001481 
.001479 
.001477 
.001475 
.001473 

.001471 
.001468 
.001466 
.001464 
.001462 
.001460 
.001458 
.001456 
.001453 
.001451 

.001449 
.001447 
.001445 
.001443 
.001441 
.001439 
.001437 



122 



MATHEMATICAL TABLES. 



No. 



Reciprocal. 



697 .001435 

698 .001433 

699 .001431 

rOO .001429 

roi .001427 

'02 .001425 

'03 .001422 

^04 .001420 

ro5 .001418 

'06 .001416 

07 .001414 

ro8 .001412 

r09 .001410 

10 .001408 

11 .001406 

12 .001404 

13 .001403 

14 .001401 

15 .001399 

16 .001397 

17 .001395 

18 .001393 

19 .001391 

'20 .001389 

'21 .001387 

22 .001385 

23 .001383 

24 .001381 

25 .001379 

26 .001377 

27 .001376 

28 .001374 

29 .001372 

^30 .001370 

'31 .001368 

'32 .001366 

'33 .001364 

'34 .001362 

'35 .001361 

36 .001359 

r37 .001357 

38 .001355 

^39 .001353 

r40 .001351 I 

'41 .001350 ! 



No. 



Reciprocal. 



742 


.001 


743 


.001 


744 


.001 


745 


.001 


746 


.001 


747 


.001 


748 


.001 


749 


.001 


750 


.001 


751 


.001 


752 


.001 


753 


.001 


754 


.001 


755 


.001 


756 


.001 


757 


.001 


758 


.001 


759 


.001 


760 


.001 


761 


.001 


762 


.001 


763 


.001 


764 


.001 


765 


.001 


766 


.001 


767 


.001 


768 


.001 


769 


.001 


770 


.001 


771 


.001 


772 


.001 


773 


.001 


774 


.001 


775 


.001 


776 


.001 


777 


.001 


778 


.001 


779 


.001 


780 


.001 


781 


.001 


782 


.001 


783 


.001 


784 


.001 


785 


.001 


786 


.001 



348 

346 

344 
342 
340 
339 
337 
335 

333 
332 

328 
326 

325 

323 
321 

319 
318 

316 

314 
312 

311 

309 

307 

305 

304 
302 

300 

299 
297 

295 
294 
292 

290 
289 
287 
285 
284 

282 
280 

279 

277 
276 

274 
272 



No. 



787 
788 

789 



830 

831 



Reciprocal. 



.001271 
.001269 
.001267 



790 


.001266 


791 


.001264 


792 


.001263 


793 


.001261 


794 


.001259 


795 


.001258 


796 


.001256 


797 


.001255 


798 


.001253 


799 


.001251 


800 


.001250 


801 


.001248 


802 


.001247 


803 


.001245 


804 


.001244 


805 


.001242 


806 


.001241 


807 


.001239 


808 


.001238 


809 


.001236 


Bid 


.001235 


811 


.001233 


812 


.001232 


813 


.001230 


814 


.001229 


815 


.001227 


816 


.001225 


817 


.001224 


818 


.001222 


819 


.OOI22I 


820 


.001220 


821 


.001218 


822 


.001217 


823 


.001215 


824 


.001214 


825 


.001212 


826 


.001211 


827 


.001209 


828 


.001208. 


829 


.001206 



.001205 
.001203 



No. 

832 

833 
834 

835 
836 

837 
838 

839 

840 

841 
842 

843 
844 

845 
846 

847 
848 

849 

850 

851 
852 

853 
854 

855 
856 

857 
858 

859 

860 

861 
862 
863 
864 
865 

866 
867 
868 
869 

870 

871 
872 

873 
874 

875 
876 



Reciprocal. 



.00: 
.00] 
.00: 
.00: 
.00: 
.00: 
.00: 
.00: 

.00: 
.00: 
.00: 
.00: 
.00; 
.00: 
.00; 
.00; 
.00; 
.00; 

.00] 
.00: 
.00] 
.00: 
.OOj 
.00: 
.00; 
.00: 
.00: 
.00: 

.00; 

.00: 

.OOJ 

.OOJ 

.00; 

.00: 

.00; 

.00: 

.00 

.00: 

.00: 

.00; 

.00: 

.00: 

.00 

.00; 

.00: 



202 
200 
199 
198 
196 

195 

193 
192 

190 
189 
188 
186 

185 
183 
182 

181 
179 
178 

176 

175 

174 
172 

171 

I 70 

168 

167 

166 

164 

^^3 
161 

160 

159 
157 
156 
155 
153 
152 

151 

149 
148 

147 

145 
144 

143 
142 



RECIPROCALS OF NUMBERS. 



123 



Now 



Rec ip io ca L 



877 


.001 


878 


.001 


879 


.OOI 


880 


.001 


S8I 


.001 


SS2 


.001 


883 


.001 


884 


.001 


885 


.001 


886 


.001 


887 


.OOI 


888 


.001 


889 


.001 


Sgo 


.001 


891 


.001 


892 


.001 


893 


.001 


894 


.001 


895 


.001 


896 


.001 


897 


.001 


898 


.001 


899 


.001 


QOO 


.001 


901 


.001 


902 


.001 


903 


.001 


904 


.001 


905 


.001 


906 


.001 


907 


.001 



140 

139 

138 

136 

135 
134 
133 
131 

130 

129 
127 
126 

125 

124 

122 
121 
120 
119 
118 
116 

"5 
114 

112 

III 
IIO 
109 
107 
106 

104 
103 



No. 



908 
909 

910 

911 
912 

914 

915 
916 

917 

918 

919 

920 
921 
922 

923 

924 

925 
926 

927 

928 

929 

931 
932 

933 
934 
935 
936 

937 
938 



Reciprocal. 



.001101 
.001100 



.001099 
.001098 
.001096 
.001095 
.001094 
.001093 
.001092 
.001091 
.001089 
.001088 

.001087 
.001086 
.001085 
.001083 
.001082 
.001081 
.001080 
.001079 
.001078 
.001076 

.001075 
.001074 
.001073 
.001072 
.001071 
.001070 
.001068 
.001067 
.001066 



No. 



939 

940 
941 

942 

943 
944 

945 
946 

947 
948 

949 

950 

951 
952 

953 
954 

955 
956 

957 

958. 

959 

960 
961 
962 

963 
964 

965 
966 

967 

968 

969 



RcoprocaL 



.001065 

.001064 
.001063 
.001062 
.001060 
.001059 
.001058 
.001057 
.001056 
.001055 
.001054 

.001053 
.001052 
.001050 
.001049 
.001048 
.001047 
.001046 
.001645 
.001044 
.001043 

.001042 
.001041 
.001040 
,001038 
.001037 
.001036 
.001035 
.001034 
.001033 
.001032 



No. 



970 

971 

972 

973 
974 

975 
976 

977 
978 

979 

980 
981 
982 

983 
984 

985 
986 

987 

988 

989 

990 

991 

992 

993 

994 

995 
996 

997 
998 

999 
1000 



Reciprocal. 



,001031 
.001030 
.001029 
.001028 
.001027 
.001026 
.001025 
.001024 
.001022 
.001021 

.001020 
.001019 
.001018 
.001017 
.001016 
.001015 
.001014 
.001013 
.001012 
.001011 

.001010 
.001009 
.001008 
.001007 
.001006 
.001005 
.001004 
.001003 
.001002 
.001001 

.001000 



WEIGHTS AND MEASURES. 



WATER AND AIR AS STANDARDS FOR WEIGHT AND MEASURE. 



WATER AS A STANDARD. 

There are four notable temperatures for water, namely, 

32° R, or 0° C. = the freezing point, under one atmosphere. 

3 9°. I or 4° = the point of maximum density. 

62° or 1 6°. 66 = the British standard temperature. 

212° or 100° = the boiling point, under one atmosphere. 

The temperature 62° F. is the temperature of water used in calculating 
the specific gravity of bodies, with respect to the gravity or density of 
water as a basis, or as unity. In France, the temperature of maximum 
density, 39°.! F., or 4° C, is used for this purpose, for solids. 

Weight of one cubic foot of Pure Water, 

At 32° F. =■ 62.418 pounds. 

At 39°. I = 62.425 „ 

At 62° (Standard temperature) = 62.355 » 
At 212° = 59.640 „ 

The weight of a cubic foot of water is, it may be added, about 1000 
ounces (exactly 998.8 ounces), at the temperature of maximum density. 

The weight of water is usually taken in round numbers, for ordinary 
calculations, at 62.4 lbs. per cubic foot, which is the weight at 52^.3 F. ; or 
it is taken at 62)^ lbs. per cubic foot, where precision is not required, equal 
to 1^^ lbs. 

The weight of a cylindrical foot of water at 62° F. is 48.973 pounds. 

Weight of one cubic inch of Pure Water, 

At 32® F. = .03612 pounds, or 0.5779 ounce. 

At39°.i =.036125 „ ,,0.5780 „ 

At 62° =.03608 „ ,,0.5773 „ or 252.595 grains. 

At 212** = .03451 „ „ 0.5522 „ 

The weight of one cylindrical inch of pure water at 62** F. is -02833 
pound, or 0.4533 ounce. 



WATER AND AIR AS STANDARDS. 1 25 

Volume of one pound of Pure Water, 

At 32** F. = .016021 cubic foot, or 27.684 cubic inches. 

At 39^1 = .016019 „ „ 27.680 „ 

At 62"* = .016037 „ „ 27.712 „ 

At 212** = .016770 „ „ 28.978 „ 

The volume of one ounce of pure water at 62** F. is 1.732 cubic inches. 

The Gallon. 

The weight of one gallon of water at the standard temperature, 62*^ F., 
is 10 pounds, and the correct volume is 0.160372 cubic foot, or 277.123 
cubic inches. But in an Act of Parliament, now partly repealed, which 
came into force in 1825, the volume of one gallon is stated to be 277.274 
cubic inches; this is the commonly accepted volume. 

The volume of 10 pounds of water at 62° F. is, therefore, to the volume 
of the imperial gallon, as i to 1.000545. 

And, the weight of an imperial gallon of water at 62° F. is 10.00545 
pounds avoirdupois j or 10 pounds 38.15 grains. 

One cubic foot of water contams 6.2355 gallons, or approximately 6^ 
gaDons. 

The volume of water at 62*" F., in cubic inches, multiplied by .00036, 
giv« the capacity in gallons. 

The capacity of one gallon is equal to one square foot, two inches deep 
nearly (exactly 1.924 inches); or to one circular foot, 2j^ inches deep 
nearly (exactly 2.45 inches). 

One ton of water at 62° F. contains 224 gallons. 

Other Measures of Water. 

Volume of given weights of water, at 62.4 pounds per cubic foot: — 

I ton 35-90 cubic feet. 

I cwt 1.795 » 

I quarter 449 „ 

, r .016 cubic foot, or 

' P^^^ • J 27.692 cubic inches. 

I ounce I-73I » 



I tonne, at 39°.! F 35-3I56 cubic feet. 

1 -1 ..or? f -0353 cubic foot, or 

I kdogiamme, at 39°.i F | 61.0^5 cubic inches. 

I tonne, at 52;.3 F. ) 35.330 cubic feet 

(62.4 pounds per cubic foot) J ^^ ^^ 

Thirty-six cubic feet, or i^ cubic yards, of water, at 62.4 pounds per 
tabic foot, being at the temperature 52^.3 F., weigh about one ton (exactly 
6.4 pounds more). 

Ctae cubic yard, or twenty-seven cubic feet, of water weighs about 
15 cwt, or ^ ton (exactly 4.8 pounds more). 

One cubic metre of water is equal in volume to 35.3156 cubic feet, 
or 1.308 cubic yard, or 220.09 gallons; and, at 62.4 pounds per cubic foot, 
it weighs i ton nearly (exactly 36.3 pounds less). It is nearly equivalent 



126 WEIGHTS AND MEASURES. 

to the old English tun of 4 hogsheads — 210 imperial gallons, and is a 
better unit for measuring sewage or water-supply than the gallon. 

The cubic metre is generally used on the Continent for such measurements. 

A pipe one yard long holds about as many pounds of water as the square 
of its diameter in inches (exactly 2 per cent. more). 

Pressure of Water. 

A pressure of one lb. per square inch is exerted by a column of water 
2.3093 feet, or 27.71 inches high, at 62® F,; and a pressure of one atmos- 
phere, or 14.7 lbs. per square inch, is exerted by a column of water 
33.947 feet high, or 10.347 metres, at 62° F. 

A column of water at 62"^ F., one foot high, presses on the base with a 
force of 0.433 lb., or 6.928 ounces per square inch. A column 100 feet 
high presses with a force of 43^ lbs. per square inch. A column one 
metre high presses with a force of 1.422 lbs. per square inch. 

A column of water one inch high, presses on the base with a force of 
0.5773 ounce per square inch, or 5.196 lbs. per square foot. 

A column of water one mile deep, weighing 62.4 pounds per cubic foot, 
presses on the base with a force of about one ton per square inch (fresh 
water exacdy 48 lbs. more; sea- water exactly 107.5 ^^s. more). 

Water is hardly compressible under pressure. Experiment appears to 
show that for each atmosphere of pressure it is condensed 47)^ millionths 
of its bulk. 

Sea-water. 

One cubic foot of average sea-water, at 62** F., weighs 64 pounds, and 
the weight of fresh water is to that of sea-water as 39 to 40, or as i to 1.026. 
Thirty-five cubic feet of sea-water weighs one ton. 
One cubic yard of sea-water weighs i^yi cwt nearly (8 lbs. less). 
One cubic metre of sea-water weighs fully one ton (20 lbs. more). 

Average sea-water is composed as follows : — 

Per xoo paits. Per zoo parts. 

Chloride of sodium • (common salt), 2.50 

Sulphuret of magnesium, 0.53 

Chloride of magnesium, 0.33 

Carbonate of lime, ) 

Carbonate of magnesia, J 

Sulphate of lime, o.oi 

Solid matter, say, 3.40 

Water, 96.60 



100.00 



» 



I 



showing that sea-water contains ^^^th part of its weight of solid matter in 
solution. 

According to R^clus, the mean specific gravity of sea-water is 1.028. In 
the Mediterranean Sea, it is 1.029; in the Black Sea, 1.016. The mean 
quantity of salts, or solid matter, in solution, is 3.44 percent., three-fourths 
of which is common salt In the Red Sea, the water contains 4.3 per cent. • 
in the Baltic Sea, 5 per cent. ; and at Cronstadt, 2 per cent 



WATER AND AIR AS STANDARDS. 12/ 

Ice and Snow, 

One cubic foot of ice at 32° F. weighs 57.50 lbs. 

One pound of ice at 32° F. has a volume of .0174 cubic foot, or 30.067 
cubic inches. 

The volume of water at 32° F. is to that of ice at 32** F., as i.ooo to 
1.0855; ^^ expansion in passing into the sohd state being above 8)^ per 
cent of the volume of water. 

The specific density of ice is 0.922, that of water at 62° F. being = i. 

The melting point of ice is ^2"^ F., or 0° C, under the ordinary atmos- 
pheric pressure, of 14.7 lbs. per square inch. Under greater pressure the 
melting point is lower, being 'at the rate of .0133° F. for each additional 
atmosphere of pressure. 

The specific heat of ice is .504, that of water being = i. 

One cubic foot of fresh snow weighs 5.20 lbs. Snow has 12 timeis the 
bulk of water, and its specific gravity is .0833. 

French and English Measures of Water, 

One litre of water is equal to 0.2201 gallon, or 1.761 pints: about 
i|:^ pints. One gallon is equal to 4.544 litres, and one pint is .568 litre. 

One litre of water at 39'^.! F., or 4° C, the temperature of maximum 
density, weighs one kilogramme, or 2.2046 lbs.; at the temperature 62° F., 
or 1 6°. 7 C, it weighs 2.202 lbs. 

looo litres = one cubic metre, equal to 35.3156 cubic feet; and, at 
39^1 F., or 4** C, weigh 1000 kilogrammes, or one ton nearly (35.4 lbs. less). 

AIR AS A STANDARD. 

The mean pressure of the atmosphere at the level of the sea, is equal 
to 14.7 lbs. per square inch, or* 21 16.4 lbs. per square foot; or to 1.0335 
kilogrammes per square centimetre. This is called one atmosphere of 
pressure. The following are measures of pressures (see also pages 1 45, 158): — 

One atmosphere of pressure : — (i.) A column of air at 32° F., 27,801 feet, 
or about 5j^ miles high, of uniform density, equal to that of air at the level 
of the sea. (2.) A column of mercury at 32° F., 29.922 inches or 76 centi- 
metres high; nearly 30 inches. At 62° F., the height is 30 inches. (3.) A 
column of water at 62** F., 33.947 feet or 10.347 metres high; nearly 34 feet. 

A pressure of i lb. per square inch: — (i.) A column of air at 32° F., 
189 1 feet high, of uniform density as above. (2.) A column of mercury at 
32' F., 2.035 inches or 51.7 millimetres high. At 62° F., the height is 2.04 
inches. (3.) A column of water at 62° F., 2.31 feet or 27.72 inches high. 

A pressure of i lb. per square foot: — (i.) A column of air at 32® F., 13.13 
feet high, of uniform density as above. (2.) A column of mercury at 32° F., 
.0141 inch or .359 millimetre high. At 62° F., the height is .01417 inch. 
(3.) A column of water at 62° F., .1925 inch high. 

The density, or weight of one cubic foot of pure air, under a pressure 
of one atmosphere, or 14.7 lbs. per square inch, is 

At 32° F., = .080728 pound, or 1.29 ounce, or 565.1 grains. 
At62**F., = .076097 „ „ 1.217 „ „ 53;2.7 „ 

The weight of a litre of pure air, under one atmosphere, at 32° F., is 
1.293 gnunmesy or 19.955 grains. 



128 WEIGHTS AND MEASURES. 

The weight of air, compared with that of water at three notable tempera- 
tures, and at 5 2°. 3, under one atmosphere, is as follows: — 

773.2 times the weight of air at 32° F. 

773-27 » » »f 

772*4 » » »  

819.4 „ „ 62^ 

820 „ „ „ 

The volume of one pound of air at 32** F., and under one atmosphere of 
pressure, is 12.387 cubic feet. The volume at 62° F., is 13. 141 cubic feet. 

The specific heat of air at constant pressure is .2377, and at constant 
volume .1688, that of water being = i. 



Weight 


of water at 


: 32° F., 






39". I, 
62°, 

62°, 


»> 


» 


Sa^-S. 



GREAT BRITAIN AND IRELAND.— IMPERIAL WEIGHTS 

AND MEASURES. 

The origin of English measures is the grain of com. Thirty-two grains 
of wheat, dried and gathered from the middle of the ear, weighed what was 
called one penn)rweight; 20 pennyweights were called one ounce, and 
20 ounces one pound. Subsequently, the pennyweight was divided into 
24 grains. Troy weight was afterwards introduced by William the Conqueror, 
from Troyes, in France; but it gave dissatisfaction, as the troy pound did 
not weigh so much as the pound then in use; consequently, a mean weight 
was established, making 16 ounces equal to one pound, and called avoir- 
dupois {avoir du poids), , 

Three grains of barleycorn, well-dried, placed end to end, made an inch 
— the basis of length. The length of the arm of King Henry I. was made 
the length of the ulna^ or ell, which answers to the modern yard. The 
imperial standard yard is a solid square bar of gun-metal, kept in the 
office of the Exchequer at Westminster, 38 inches in length, i inch square, 
at the temperature 62° F., composed of copper 16 ounces, tin 2^ ounces, 
and zinc i ounce. Two cylindrical holes are drilled half through the bar, 
one near each end, and the centres of these holes are 36 inches, or 3 feet, 
apart — the length of the imperial standard yard. Compared with a pendu- 
lum vibrating seconds of mean time, at the level of the sea, in the latitude 
of London, in a vacuum, the yard is as 36 inches in length to 39.1393 
inches, the length of the pendulum. 

Measures of capacity were based on troy weight; it was enacted that 
8 pounds troy of wheat, from the middle of the ear, well dried, should 
make i gallon of wine measure, and that 8 such gallons should make 
I bushel. 

The imperial gallon is now the only standard measure of capacity, and it 
contains 277.274 cubic inches. It is said to be the volume of 10 pounds 
avoirdupois of distilled water, weighed in air, at 62** F. 

Note, — The exact volume of 10 pounds of distilled water at 62** F. is 
277.123 cubic inches. 



GREAT BRITAIN AND IRELAND.— LENGTH. 1 29 

Tables of weights and measures are conveniently classified thus — 
I. Length; 2. Surface; 3. Volume; 4. Capacity; 5. Weight. 

The following are some of the principal units of measurement : — 

The acr^j for land measure. 

The mi/^, for itinerary measure. 

The yardj for measure of drapery, &c. 

The coomb^ for capacity of com, &c. 

The gallon^ for capacity of liquids. 

The graifij for chemical analysis. 

TYsQ found, for grocers* ware, &c 

The stone of 8 pounds, for butchers' meat 

The stone of 14 pounds, for flour, oatmeal, &c. 

I. Measures of Length. — Tables No. 12. 

Lineal Measure. 

3 barleycorns, or'V 

12 lines, or f • i. 

72 pomts, or I 

1000 mils / 

3 inches i palm. 

4 inches i hand. 

9 inches v i span. 

12 inches '. i foot 

18 inches i cubit 

3 feet I yard. 

2^ feet i.military pace. 

5 feet I geometrical pace. 

2 yards i fathom. 

5j^ yards * i rod, pole, or perch. 

zt^r] ^f-i-«- 

8 furlongs, or "j 

1760 yards, or > i mile. 

5280 feet j 

3 miles I league. 

2240 yards, or ) j . , ., 



The inch is also divided into halves, quarters, eighths, and sixteenths; 
sometimes mto tenths. 

The hand is used as a measure of the height of horses. 

The miUtary face is the length of the ordinary step of a man. 

The geometrical pace is the length of two steps. A thousand of such 
paces were reckoned to a mile. 

The fathom is used in soundings to ascertain depths, and for measuring 

cordage and chains. 

9 



130 WEIGHTS AND MEASURES. 

Land Measure, 

7.92 inches i link. 

100 links, or \ 

^^^^^\^^ \ X chain. 

22 yards, or f 

4 poles ) 

10 chains i furlong. 

80 chains, or ) _., 

8 furlong / ^ °^^ 

The^, or woodland pole ox ferchy is 18 feet 
Tht forest pole \s 21 feet. . 

Nautical Measure. 

6086.44 feet, or \ 

1000 fathoms, or ( f i nautical mile, 

10 cables, or C ( or knot. 

1. 1528 statute miles ) 

3 nautical miles i league. 

60 nautical miles, or \ 

69.168 statute miles or > i degree. 

20 leagues j 

( Circumference 

360 degrees...' < of the earth at 

( the equator. 

The above value of the nautical mile is that which is commonly taken, 
and is the length of a minute of longitude at the equator. The mean 
length of a minute of latitude at the mean level of the sea is nearly 6076 
feet, or 1.1508 statute miles. 

The nautical fathom is the thousandth part of a nautical mile, and is, on 
an average, about ^th longer than the common fathom. 

Cloth Measure, 

2}^ inches , i nail. 

2 nails I finger-length. 

4 nails, or 9 inches i quarter. 

4 quarters i yard. 

5 quarters i elL 

Wire-Gauges. 

The " Birmingham Wire-Gauge " is a scale of notches in the edge of a 
plate, of successively increasing or decreasing widths, to designate a set of 
arbitrary sizes or diameters of wire, ranging from about half an inch down to 
the smallest size easily drawn, say, four-thousands of an inch. The practical 
utility of such a gauge is obvious, when it is considered how far beyond the 
means supplied by the graduations of an ordinary scale of feet and inches 
is the measurement of the gradations of the wire-gauge. But the "Birming- 
ham Wire-Gauge" is a variable measure. The principle, if there was any, 
on which it was originally constructed, is not known. Mr. Latimer Clark 
states that, when plotted, the width? of the gauge range in a curve approxi- 



GREAT BRITAIN AND IRELAND. — WIRE-GAUQES. 



131 



mating to a logariAmic curve, such as would be found by the successive 
addition of 10 or 12 per cent to the width of the notches of the gauge. 
However that may be, there are many varieties of the wire-gauge in existence. 
The oldest and best-known gauge is that of which the numbers were care- 
fully measured by Mr. Holtzapffel, and published by him in 1847. It has 
been, and still is, widely followed in the manufacture of wire; and also of 
tubes in respect of their thickness. It gives 40 measurements ranging from 
.454 inch to .004 inch, and is contained in Table No. 13. Although 
there are only 40 marks in the table, there are 60 different sizes of wire 
made, for which intermediate sizes have been added to the gauge. This 
table haJs also been used in rolling sheet iron, sheet steel, and other 
materials, and for joiners' screws; but it appears to be falling into disuse 
for these purposes. 

Birmingham Wire-Gauge (HoltzapffeVs), — Table No. 13. 
For Wire and Tubes chiefly; and for Sheet Iron and Steel formerly. 



! Maik. 


SlTT, 


Mark. 


Si2e. 


Mark. 


Size. 


Mark. 


Size. 


No. 


Inch. 


No. 


, Inch. 


No. 


Inch. 


No. 


Inch. 


0000 


.454 


7 


.180 


17 


.058 


27 


.016 


000 


.425 


8 


.165 


18 


.049 


28 


.014 


00 


.380 


9 


.148 


19 


.042 


29 


.013 





•340 


10 


.134 


20 


•035 


30 


.012 


I 


.300 


II 


.120 


21 


.032 


31 


.010 


2 


.284 


12 


.109 


22 


.028 


32 


.009 


3 


-259 


13 


•09s 


23 


.025 


33 


.008 


4 


.238 


14 


.083 


24 


.022 


34 


.007 


5 


.220 


15 


.072 


25 


.020 


35 


.005 


6 


.203 


16 


.065 


26 


.018 


36 


.004 



BiRMiNGH.^M Metai^Gauge, or Plate-Gauge (HoltzapffePs), — 

Table No. 14. 

For Sheet Metals, Brass, Gold, Silver, &c. 



1 M«k. 


Six. 


Mark. 


Sire. 


Mark. 


Size. 


Mark. 


Size. 


Ko. 


Inch. 


No. 


Inch. 


No. 


.Inch. 


No. 


Inch. 


I 


.004 


10 


.024 


19 


.064 


28 


.120 


2 


.005 


II 


.029 


20 


.067 


29 


.124 


3 


.008 


12 


.034 


21 


.072 


30 


.126 


4 


.010 


13 


.036 


22 


.074 


31 


.133 


5 


.012 


14 


.041 


23 


.077 


32 


.143 


6 


.013 


15 


.047 


24 


.082 


33 


.145 


7 


'OI5 


16 


•051 


25 


.095 


34 


.148 


8 


.016 


17 


.057 


26 


.103 


35 


.158 


9 


.019 


18 


.061 


27 


.113 


36 


.167 



Another of HoltzapffeVs tables, No. 14, the Plate-Gauge^ has been, and 
may now, to some extent, be, employed for most of the sheet metals, except- 



132 



WEIGHTS AND MEASURES. 



Lancashire Gauge {Holtzapffd' s\ — Table'^o. 15. 
For Round Steel Wire, and for Pinion Wire. 



Maik. 
No. 


Size. 


Mark. 

No. 


Size. 


Mark. 


Sirr. 


Mark. 

No. 


Size. 


Maik. 


Size. 


Inch. 


Inch. 


No. 


Inch. 


Inch. 


No. 


Inch. 


80 


.013 


57 


.042 


34 


.109 


II 


.189 


M 


•295 


79 


.014 


56 


.044 


33 


.III 


10 


.190 


N 


.302 


78 


.015 


55 


.050 


32 


•115 


9 


.191 





.316 


77 


.016 


54 


•055 


31 


.118 


8 


.192 


P 


.323 


76 


.018 


53 


.058 


30 


•125 


7 


.195 


Q 


•332 


75 


.019 


52 


.060 


29 


.134 


6 


.198 


R 


•339 


74 


.022 


51 


.064 


28 


.138 


5 


.201 


S 


•348 


73 


.023 


50 


.067 


27 


.141 


4 


.204 


T 


.358 


72 


.024 


49 


.070 


26 


■143 


3 


.209 


U 


.368 


71 


.026 


48 


•073 


25 


.146 


2 


.219 


V 


377 


70 


.027 


47 


.076 


24 


.148 


I 


.227 


w 


.386 


69 


.029 


46 


.078 


23 


.150 


A 


.234 


X 


■397 


68 


.030 


45 


.080 


22 


.152 


B 


.238 


Y 


.404 


67 


.031 


44 


.084 


21 


.157 


C 


.242 


Z 


-413 


66 


.032 


43 


.086 


20 


.160 


D 


.246 


Ai 


.420 


65 


*^ZZ 


42 


.091 


19 


.164 


E 


.250 


Bi 


■431 


64 


.034 


41 


.095 


18 


.167 


F 


•257' 


Ci 


•443 


63 


•035 


40 


.096 


17 


.169 


G 


.261 


Di 


.452 


62 


.036 


39 


.098 


16 


.174 


H 


.266 


Ei 


.462 


61 


.038 


38 


.100 


15 


.175 


I 


.272 


Fi 


.475 


60 


.039 


37 


.102 


14 


.177 


J 


.277 


Gi 


■484 


59 


.040 


36 


•105 


13 


.180 


K 


.281 


Hi 


.494 


58 


.041 


35 


.107 


12 


.185 


L 


.290 







ing iron and steel : as copper, brass, gilding-metal, gold, silver, and platinum. 
The intervals are closer or smaller than those of the wire-gauge, and the 
maximum size, for No. 36, is '/6 inch. When thicker sheets are wanted, 
their measures are sought in the Birmingham wire-gauge. 

The last table, No. 15, by Holtzapffel, the Lancashire Gauge, is employed 
exclusively for the bright steel wire prepared in Lancashire, and the steel 
pinion-wire for watch and clock makers. The larger sizes are marked by- 
capital letters, to distinguish them from the others. This, the second part 
of the table, is known as the Letter-Gauge, 

Needle- Gauge, for needle wire. The sizes correspond with some of those 
of the Holtzapffel wire-gauge. The following are the relative marks for 
equal sizes on the two gauges : — 

Needle wire -gauge — Nos. i, 2, 2j^, 3, 4, 5, thence to 21, 
corresponding to B. W.-G. — i8j^, 19, 19^, 20, 21, 22, thence to 38. 

* Music IVire-gauge, for the strings of pianofortes. The marks used are 
Nos. 6 to 20. The following are the relative marks for equal sizes with the 
Holtzapffel wire-gauge: — 

Music wire-gauge — Nos. 6, 7, 8, 9, 10,11, 12,14,16,18,20, 
corresponding to B. W.-G. — 26, 25)^, 25, 24}^, 24, 23 J^, 23, 22, 21, 20, 19. 
No. 6, the thinnest wire now used, measures about one fifty-fifth of an inch 
in diameter, and No. 20 about one twenty-fifth of an inch. 



GREAT BRITAIN AND IRELAND. — WIRE-GAUGES. 



133 



The preceding Tables of Gauges have been extracted from HoltzapfFers 
estimable work on Turning and Mechanical Majnipulation^ 1847. 

Messrs. Rylands Brothers, of Warrington, manufacture iron wire accord- 
iDg to the gauge in Table No. 16. 

Warrington Wire-Gauge (Rylands Brothers), — Table No. 16. 



Mark. 


Size. 


Mark. 


Size. 


Mark. 


Size. 


Mark. 

No. 


Size. 


No. 


Inch. 


No. 


Inch. 


No. 


Inch. 


Inch. 


7/0 


1/2 





.326 


8 


•159 


15 


.069 


6/0 


15/32 


I 


.300 


9 


.146 


16 


.0625, or Vx6 


5/0 


7/16 


2 


.274 


TO 


•133 


17 


.053 


4,^ 


13/32 


3 


-25, or K 


io>^ 


.125, or^ 


18 


.047 


3/0 


3/8 


4 


.229 


II 


.117 


19 


.041 


2/0 


11/32 


5 


.209 


12 


.io,or V,o 


20 


.036 






6 


.191 


13 


.090 


21 


•o3i5»orV33 






7 


•174 


14 


•079 


22 


.028 



For sheets, the wire-gauge that seems to be adhered to by the iron-sheet 
rollers of South Staffordshire, is a scale comprising 32 measurements, ranging 
from .3125 inch to .0125 inch, contained in Table No. 17. 

Birmingham Wire-Gauge. — ^Table No. 17. 
For Iron Sheets chiefly. 



No. 



Size. 


No. 


Inch. 


3125 (Vx6) 


9 


28125 


10 


25 (X) 


II 


234375 


12 


21875 


13 


203125 


14 


1875 (V,6) 


15 


171875 


16 



Size. 


No. 


Inch. 




•15625 
.140625 
.125 (>^) 


17 
18 

19 


.1125 


20 


•10 (Vio) 
.0875 


21 
22 


.075 

.0625 (Vxe) 


23 

24 



Size. 


No. 


Inch. 


.05625 


25 


.05 (V-o) 


26 


•04375 


27 


.0375 


28 


•034375 


29 


.03125(732) 


30 


.028125 


31 


•025 (V40) 


32 



Inch. 
.02344 
.021875 
.020312 
.01875 
.01719 
.015625 
.01406 

.0125 (Veo) 



Sir Joseph Whitworth, in 1857, introduced his Standard Wire-Gauge, 
ranging fix)m a half inch to a thousandth of an inch, and comprising 62 
measurements, as given in Table No. 18. It commences with the 
smallest size, and increases by thousandths of an inch up to half an inch. 
The smallest size, V'loooth of an inch, is No. i ; No. 2 is Vioooths of an inch, 
and so on, increasing up to No. 20 by intervals of Vioooth of an inch; from 
No. 20 to No. 40 by '/loooths; from No. 40 to No. 100 by s/ioooths of an 
inch. The sizes are designated or marked by their respective numbers in 
thousandths of an inch. 

It appears that the Whitworth Gauge is entering into general use ; and, 
in the manufacture of wire, at least, this and Rylands* gauge are likely soon 
to supersede the Holtzapifel scale. 



134 



WEIGHTS AND MEASURES. 



Sir Joseph Whitworth & Co.'s Standard Wire-Gauge. — ^Table No. i8. 



Mark. 


Size. 


Mark. 


Size. 


Mark. 


Size. 


Mark. 


Size. 


No. 


Inch. 


No. 


Inch. 


No. 


Inch. 


No. 


Inch. 


I 


.001 


17 


.017 


55 


.055 


200 


.200 


2 


.002 


18 


.018 


60 


.060 


220 


.220 


3 


.003 


19 


.019 


6$ 


.065 


240 


.240 


4 


.004 


20 


.020 


70 


.070 


260 


.260 


S 


.005 


22 


.022 


75 


.075 


280 


.280 


6 


.006 


24 


.024 • 


80 


.080 


300 


.300 


7 


.007 


26 


.026 


85 


.085 


325 


.325 


8 


.008 


28 


.028 


90 


.090 


350 


.350 


9 


.009 


30 


.030 


95 


.095 


375 


•375 


ID 


.010 


32 


.032 


100 


.100 


400 


.400 


II 


.Oil 


34 


.034 


no 


.110 


425 


.425 


12 


.012 


36 


.036 


120 


.120 


45«> 


.450 


13 


.013 


3S 


.038 


135 


•135 


475 


.475 


14 


.014 


40 


.040 


150 


.150 


500 


.500 


15 


.015 


45 


.045 


165 


.165 






16 


.016 


50 


.050 


180 


.180 







Common Fractions of an Inch and Holtzapffel's Wire-Gauge. — 

Table No. 19. 



Fraction. 



Inch. 

'A 
'A 
■A 
'A 

'A 
•A 

■A 

'Ac 

'A. 

•A, 

'/.6 
•/ 

:^ 



Wire-Gauge. 



so 



21 



a« 



24 



25 
Va6 



No. 

fuU 

bare 

bare 

bare 

full 

full 

full 

bare 



o 

3 
6 

8 

9 
I 

2 

3 
3 
4 

4 bare 

5 rather bare 
6 

6 bare 
7fuU 

7 bare 

8 full 

8 rather full 
8 rather bare 

8 bare 

9 rather full 

9 

9 bare 

20 full 



Fraction. 



Inch. 
Va8 

V3« 
V37 
V38 
V39 

'Ac 

v:: 

'r 

Vs6 
V58 
V60 



Wire-Gauge. 



No. 

20 rather full 
20 

20 rather bare 

21 rather full 
21 

21 rather bare 

21 bare 

22 full 

22 rather full 
22 

22 bare 

23 full 

23 rather full 

23 

23 bare 

24 full 

24 
25fuU 

25 

25 bare 

26 rather full 
26 

26 bare 

27 full 



Fraction. 



Wire-Gauge. 



Inch. 



V 



70 

V74 
V76 

'As 

Vso 

•As 

:^ 

/95 

V 

:;■ 

Vx3o 
V140 

Vis© 

VI60 

180 



100 



zo 



xao 



V 



7 



aoo 



950 



No. 
27 

27 rather 

27 bare 

28 full 
28 rather 
28 

28 bare 

29 rather 
29 

29 bare 

30 rather 

30 bare 

31 rather 

31 
32 
33 full 

33 l>are 

34 

34 rather 

34 bare 

35 Ml 
35 

36 



bare 

full 

full 

full 
full 



bare 






GREAT BRITAIN AND IRELAND. — FRACTIONS OF INCH. 1 35 

Inches and their Equivalent Decimal Values in Parts of a Foot. 

— Table No. 20. 



Inches. 


Fiactionoffoot 


Foot. 


I 


Via 


08^-1 


2 

7 


:; 


.1667 
25 




4 

c 


1 


•3333 
4167 


D 

6 

7 


■A 

7/„ 


...... .Af * vr f 

■5 
eS-iX 


/ 

8 



/la 

Va 
3/. 


oo 

.6667 

7c 


V 

10 

II 




/o 

.8333 

QI67 


12 

• 


/ xa 

I 


y *^/ 

I.O 



Fractional Parts of an Inch, and their Decimal Equivalents. 

Tables No. 21. 

Eighths. 



Eighths. 


Fractions. 


Inch. 


I 


i/g 


12^ 


2 

X 


/o 

•A 

S/g 


^•J 

77c 




4 

c 


/<» 

V. 

S/g 


o# J 

•5 
621: 




6 

7 


/«» 

7/. 


•*'• J 

•75 
87s 


/ •••••• 

8 


I 


/ J 

1.0 



Twelfths, 



Twelfths. 


Fractions. 


Inch. 


I...... 


«/„ 


oZx%% 


2 


'A 
i/« 


•125 

16667 


3 
4. 


'A 

V, 


•^5 
'?'?^^'? 


^ 

5 

6. 


'k 


•• 00000 

.41667 
5 


7 

8 


/* 

t 


•5^333 
66666 


9 
10 


5/g 


•75 
S'?^^^ 


II 

12 


/o 

"A. 

I 


^0000 

.91667 
1.0 









136 WEIGHTS AND MEASURES. 

Sixteenths and Thirty-seconds. — Tables No. 21 (continued). 



Thirty- 
Seconds. 


Sutteenths. 


Fractions. 


Inch. 


I 
2 

3 

4 

5 
6 

7 
8 

9 
10 

II 

12 

13 

14 

IS 
16 

17 
18 

19 
20 

21 

22 

23 
24 

25 
26 

27 
28 

29 
30 
31 
32 


I 

2 

3 

4 

5 
6 

7 
8 

9 
10 

II 

12 

13 
14 

15 
16 


v„ .: 


.03125 
.0625 

•0937s 

"5 

•15625 

.1875 
.21875 

•*S 
.28125 

.3125 

•34375 

•375 
.40625 

•4375 
.46875 

•S 

•53125 

•5625 

•59375 
.625 

.65625 

•6875 

•71875 

•75 
.78125 

.8125 

.84375 

.875 
.90625 

•9375 
.96875 

1.0 


/3* 

3/,. 


/ 3' 

5/„ 


'3a • 
7/„ 


/3« 

V, :'•.... 




/3* 

3/3 

»3/„ 


/3a 

»/. !{:!... 


»7/„ 


/ 3* 

'A6 
«9/^ 


/3« • 

91/ 


/ 3* 

"A6 
23/„ 


/ 3« 

»s/.. 


'3« • 
a7/„ 


/ 3* 

ap/,^ 


'3* • 

'V.6 
3x/„ 


/ 3* 

I 



II. Measures of Surface. — Tables No. 22. 

Superficial Measure, 

144 square inches, or 7 - 

183.35 circular inches 3 i square foot 

9 square feet i square yard 

loo square feet i square. 

27 2 J^ square feet, or ) nH 

30^ square yards j ' ^^* 

The square is used in measuring flooring and roofing. 
The rod is used in measuring brick-work. 



GREAT BRITAIN AND IRELAND. — SURFACE, VOLUME. 1 37 

Builder^ Measurement, 

I superficial part i square inch. 

12 parts "i inch" (12 square inches). 

12 "inches" i square foot. 

This table is employed in the superficial or flat measure of boards, glass, 
stone, artificers' work, &c. 

Land Measure. 

9 square feet i square yard. 

30X square yards -j ^ ^^l^^ 

16 square poles i square chain. 

40 square poles, or ) ^ 

1 2 10 square yards J 
4 roods, or 
10 square chains, or 

160 square poles, or \ i acre.* 

4,840 square yards, or 
43,560 square feet 

640 acres, or ) -1 

3,097,600 squa^ yards } i square imle. 

30 acres i yard of land. 

100 acres i hide of land. 

40 hides I barony. 

* The side of a square having an area of one acre is eqnal to 69.57 lineal yards. 

III. Measures of Volume. — Tables No. 24. 

Solid or Cubic Measure, 

1728 cubic inches \ 

2200.15 cylindrical inches ( i cubic foot 

3300.23 spherical inches j 
6600.45 conical inches ) 

27 cubic feet i cubic yard, or load. 

35.3156 cubic feet or \ ^ ^^^j^ 

1.308 cubic yards J 

^ate. — ^The numbers of cylindrical, spherical, and conical inches in a cubic foot, are 
as I, 1. 5, 3- 

Builderi Measurement, 

I solid part 12 cubic inches. 

12 solid parts i "inch" (144 cubic inches). 

12 "inches" i cubic foot 

This table is used in measuring square-sided timber, stone, &c. 



138 



WEIGHTS AND MEASURES. 



Note, — ^The cubic contents of a piece, 

6 inches square and 4 feet long is i cubic foot 



7 

8J^ 
12 

17 
24 






3 

2 

I 
I 
I 



» 



I 
I 
I 
2 

4 



Decimal Parts of a Square FogTj 


, IN Square Inches. — Table No. 23. 


Hundredth 


Square 


Hundredth 


Square 


Hundredth 


Square 


Hundredth 


Square 


Parts. 


Inches. 


Parts. 


Ihches. 


Parts. 


Inches. 


Parts. 


Inches. 


I 


1.44 


26 


37.4 


51 


73.4 


76 


109.4 


2 


2.88 


27 


38.9 


52 


74.9 


77 


IIO.9 


3 


4.32 


28 


40.3 


53 • 


76.3 


78 


II2.3 


4 


5-76 


29 


41.8 


54 


77.8 


79 


1 13.8 


5 


7.20 


30 


43.2 


55 


79.2 


80 


II5.2 


6 


8.64 


31 


44.6 


56 


80.6 


81 


I16.6 


7 


10. 1 


32 


46.1 


57 


82.1 


82 


1 18. 1 


8 


"5 


33 


47.5 


58 


83.5 


83 


"95 


9 


13.0 


34 


49.0 


59 


85.0 


84 


121.0 


10 


14.4 


35 


50-4 


60 


86:4 


85 


122.4 


II 


iS-8 


36 


51.8 


61 


87.8 


86 


123.8 


12 


17.3 


37 


53-3 


62 


89.3 


87 


125.3 


13 


18.7 


38 


54.7 


63 


90.7 


88 


126.7 


14 


20.2 


39 


56.2 


64 


92.2 


89 


128.2 


. 15 


21.6 


40 


57.6 


65 


93-6 


90 


129.6 


16 


23.0 


41 


58.0 


66 


95.0 


91 


131.0 


17 


24.5 


42 


60.5 


67 


96.5 


92 


132.5 


18 


259 


43 


61.9 


68 


97.9 


93 


133.9 


19 


27.4 


44 


63.4 


69 


99-4 


94 


135-4 


20 


28.8 


45 


64.8 


70 


100.8 


95 


136.8 


21 


30.2 


46 


66.2 


71 


102.2 


96 


138.2 


22 


317 


47 


67.7 


72 


103.7 


97 


139-7 


23 


33.1 


48 


69.1 


73 


105.1 


98. 


141. 1 


24 


34.6 


49 


70.6 


74 


106.6 


99 


142.6 


25 


36.0 


50 


72.0 


75 


108.0 


100 144.0 



IV. Measures of Capacity. — Tables No. 25. 

Liquid Measure, 

8.665 cubic inches i gill or quartern. 

4 gills (34.659 cubic inches) i pint 

2 pints I quart 

2 quarts i pottle. 

4 quarts, or 8 pints (277.274 cubic inches) i gallon. 

6.2355 gallons I cubic foot 

The ham-gallon^ for milk, is equal to 2 imperial gallons. 



GREAT BRITAIN AND IRELAND. — CAPACITY. 1 39 

Dry Measure, 

2 pints I quart 

4quarts i gallon. 

2 gallons I peck. 

8S;s°'} (I- 28366 cubic feet) i bushel 

2 bushels I strike. 

4bushels i coomb. 

5 bushels I sack. 

8 bushels i quarter. 

4 quarters (41.Q77 cubic feet) i chaldron. 

5quarters i wey or load. 

2 loads I last 

The standard bushel is 18^ inches in diameter inside, and 8^ inches 
deep; it holds 80 pounds of distilled water at 62'' F. It is 19^^ inches in 
diameter outside. This measure is applied to dry goods, as com, seeds, 
roots, &C., which are struck with a straight bar. The old dry measures had 
the same denominations and proportions, and were 96.95 per cent of the 
Imperial Dry Measures, above given. 

The heaped imperial bushel must be an upright cylinder, of which the 
diameter is not less than twice its depth, and the height of the conical heap 
must be at least three-fourths of the depth of the bushel, the outside of the 
pleasure being the boundary of the base of the cone. It may be 18.789 
inches in diameter inside, and 8 inches deep; and the capacity, heaped, 
must be 1.6293 cubic feet. Heaped measure is used for such goods as 
camiot be conveniently stricken, as coals, potatoes, and fruit 

Coal Measure, 

3 bushels (heaped) i sack. 

9 bushels I vat 

36 bushels, or 1 2 sacks i chaldron. 

5^ chaldrons i room. 

21 chaldrons i score. 

Old Wine and Spirit Measure, 

•11 \ Imperial 

4 g^lS I pint Gafions. 

2 pints I quart 

4 quarts (231 cubic inches) i gallon = .8333 

10 ^lons ranker = 8.333 

18 gallons I runlet = 15. 

31}^ gallons I barrel = 26.250 

42 gallons I tierce = 35. 

'jgS-"'}  hogshead =5.5 

.^*CS^°J} , p»,cheo„ . 70. ; 

126 gallons, or \ 

2 hogsheads, or > i pipeorbutt= 105. 

i^ puncheons ) 

2 pipes, or ) ^ .^ -^,r. 
*^ '^ i > I tun =210. 

3 puncheons j 



I40 WEIGHTS AND MEASURES. 

By this measure wines, spirits, cider, perry, mead, vinegar, oil, &c., are 
measured; but the contents of every cask are reckoned in imperial gallons 
when sold. The imperial gallon is one-fifth laiger than the old wine 
gallon. 

Old Ak and Beer Measure, 

2 pints I quart JSESS!* 

4 quarts (282 cubic inches) i gallon = 1.017 

9 gallons I firkin = 9.153 

2 firkins, or 18 gallons 1 kilderkin = 18.306 

2 kilderkins, or I , u^^^i r a 

, 11 ' > I barrel = 36.612 

36 gallons J ^ 

zoiX'"} ^''"" =^°^-«^^ 

The imperial gallon is one-sixtieth smaller than the old beer gallon. 

Apothecaries Fluid Measure, 

60 minims (ni) i fluid drachm (/5). 

8 drachms (water, 1.732- cubic) « • , / ^ ^x 

inches, is? >^ grains) } ^ ^^'^ ^"""^ (/ D- 
20 ounces i pint ( ^ )• 

8 pints (water, 70,000 grains) i gallon {i^^-)- 

1 drop I grain. 

60 drops I drachm. 

4drachms i tablespoonfiiL 

2 ounces (water, 875 grains) i wineglassful. 

3 ounces i teacupful. 

V. Measures of Weight. — Tables No. 26. 

Avoirdupois Weight, 

16 drachms, or I / v 

,^^1/ ,^:„« \ I ounce(^2:.). 

437?^ grams j • ^ ' 

16 ounces, or ) j /• • i\ //z \ 

7000 grains } ^ P^""*' (""Penal) {ib.). 

8 pounds I stone (London meat market). 

14 pounds I stone. 

28 pounds, or I ^ / \ 

2 stones I ' quarter (^r.). 

4 quarters, or \ 

8 stones, or > i hundredweight {cwt). 

112 pounds j 
20 hundredweights i ton. 

The grain above noted, of which there are 7000 to the pound avoirdupois, 
is the same as the troy grain, of which there are 5760 to the troy pound. 

Hence the troy pound is to the avoirdupois pound as i to 1.2 15, or as 
14 to 17. 



\ 



GREAT BRITAIN AND IRELAND.— WEIGHTS. I41 

The troy ounce is to the avoirdupois ounce as 480 grains, the weight of 
the former, to 43 7 >^ grains, the weight of the latter; or, as 1 to .9115. 
In Wales, the iron ton is 20 cwt of 120 lbs. each. 

Troy Weight. 

24 grams i pennyweight (^/o^/.). 

20 pennyweights, or ) 

480 grains / ' ^^"^^• 

1 2 ounces, or ) • 

5760 grains / ^ P°"°<^ 

25 pounds I quarter. 

4 quarters, or loo poUnds i hundredweight. 

By troy weight are weighed gold, silver, jewels, and such liquors as are 
sold by weight 

Diamond Weight 

I diamond grain 0.8 troy grain. 

I carat 4 diamond grains. 

15J4 carats i troy ounce. 

Apothecaries Weight. 

The revised table of weights of the British Pharmacopeia is as follows : 
it is according to the avoirdupois scale : — 

4375^ grains.... i ounce. 

16 ounces i pound. 

In the old table of Apothecaries* Weight, superseded by the preceding 
table, the troy scale was followed, thus: — 

Old Apothecaries Weight. 

20 grains i scruple O). 

3 scruples, or ) j 1. /-\ 

6ograi^ I I drachm (3). 

8 drachms, or ) ' /«\ 

480 grains } ^ °"'»^« <?)• 

12 ounces, or ) j /7il\ 

5760 grains } • ^ POund (/*.)• 

Weights of Current Coins. 

I farthing, .8 inch diameter, »/,o ounce. 

I halfpenny, i.o „ Yj „ 

I penny, 1.2 „ V3 » 

I threepenny piece »/«> ^i 

I fourpenny piece '/xs „ 

I sixpence Yio „ 

I shilling V5 „ 

I florin »/s » 

I half-crown Ya „ 

5 shillings or 10 sixpences i „ 

I sovereign »/^ ounce, nearly. 

For the exact weight in grains of these coins, see Table of British Money. 



142 WEIGHTS AND MEASURES. 

Coal Weight. 

14 pounds I stone. 

28 pounds I quarter hundrpdweighL 

56 pounds I half hundredweight. 

88 pounds i bushel.* 

I sack, of 1 1 2 pounds i hundredweight 

I double sack, of 224 pounds... 2 hundredweights. 

20 hundredweights, or I , 
10 double sacks J 

26j^ hundredweights i chaldron (London). 

53 hundredweights i chaldron ( Newcasde). 

7 tons T room. 

21 tons 4 cwt I barge or keel. 

* Sundry Bushel Measures, 

I Cornish bushel of coal is 90 or 94 pounds ;. heaped, loi pounds. 
I Welsh bushel, average wei£[ht 93 pounds. 

I Newcastle bushel is 80 or 84 pounids. Bradley Main, 92^ pounds. 
I London bushel, 80 or 84 pounds. 

f In Wales the miners* coal-ton is 21 cwt. of 120 lbs. each. 

Wool Weight 

' 7 pounds I clove. 

2 cloves, or 14 pounds i stone. 

2 stones I tod. 

6]4 tods I wey. 

2 weys I sack. 

12 sacks, or 39 hundredweight i last. 

12 score, or 240 pounds i pack. 

Hay and Straw Weig/U. 

I truss of straw 36 pounds. 

I load of straw 11 hundredweights, 64 pounds. 

I truss of old hay 56 pounds. 

I load of old hay 18 hundredweight. 

I cubic yard of old hay 9 stone. 

I cubic yard of oldish hay 8 stone. 

I truss of new hay 60 pounds. 

1 load of new hay ; 19 hundredweights, 3 2 pounds. 

I cubic yard of new hay 6 stone. 

Com and Flour Weight 

1 peck, or stone of flour 14 pounds. 

10 pecks I boll = 140 

2 bolls I sack =280 

14 pecks I barrel =196 

I bushel of wheat 60 

I bushel of barley ^ 47 

I bushel of oats 40 

Six bushels of wheat should yield one sack of flour; i last of corn is 80 bushel& 



GREAT BRITAIN AND IRELAND.— MISCELLANEOUS. I43 

Miscellaneous Tables. — No. 27. 

Whatman* s Drawing Papers, — Sizes of Sheets, 

Antiquarian 53 inches long, 3 1 inches wide. 

Double-elephant 40 

Atlas 34 

Colombier 34 

Imperial 30 

Elephant 28 

Super-royal 27 

Royal 23 

Medium 22 

Demy 20 



n 



27 




26 




23 




22 




23 




19 




19 




17 




15 





Commercial Numbers and Stationery. 

12 articles 1 dozen. 

13 articles i long dozen. 

12 dozen i gross. 

20 articles i score. 

5 score I common hundred. 

6 score i great hundred. 

30 deals I quarter. 

4 quarters i hundred. 

24 sheets of paper i quire. 

20 quires i ream. 

2ij| quires i printers* ream. 

5 dozen skins of parchment i roll. 

Measures relating to Building, 

Load of timber, unhewn or rough 40 cubic feet 

Load, hewn or squared [ 5° cubic feet, reckoned 

* ^ (to weigh 20 cwt 

Stack of wood io8 cubic feet. 

Cord of wood 128 „ 

(In dockyards, 40 cubic feet of hewn timber are reckoned to weigh 

20 cwt. ; 50 cubic feet is a load.) 

100 superficial feet i square. 

Himdred of deals 120 deals. 

Load of i-inch plank 600 square feet. 

(Load of plank more than i-inch thick = 600 -^ thickness in inches. 

Planks, section 11 by 3 inches. 

Deals, section 9 by 3 „ 

Battens, section 7 by 2^ „ 

A reduced deal is 1 J^ inches thick, 1 1 inches wide, and 1 2 feet long. 

Bundle of 4 feet oak-heart laths 120 laths. 

Load of „ „ 3 7 J^ bundles. 

Bundle of 5 feet oak-heart laths 100 laths. 

Load of „ „ 30 bundles. 



144 WEIGHTS AND MEASURES. 

Measures reiaiing to Building {continued,) 

Load of statute bricks 500. 

Load of plain tiles 1000, 

Load of lime 32 bushels. 

Load of sand 36 „ 

Hundred of lime 35 „ 

Hundred of nails, or tacks 120. 

Thousand of nails, or tacks 1 200. 

Fodder of lead iqJ^ cwt. 

Sheet lead 6 to 10 pounds per sq, ft 

Hundred of lead 112 pounds. 

Table of glass 5 feet. 

Case of glass 45 tables. 

case of glass { ^^^"^^f taS~'' 

Stone of glass 5 pounds. 

Seam of glass 24 stone. 

Sundry Commercial Measures, 

Dicker of hides ; 10 skins. 

Last of hides 20 dickers. 

Weigh of cheese 256 pounds. 

Barrel of herrings 26 V3 gallons. 

Cran of herrings 37^ „ 

Pocket of hops i ^ to 2 cwt. 

Bag of hops 3j^ cwt, nearly. 

Last of potash, cod-fish, white her- ) barrels. 

rings, meal, pitch, tar j 

Barrel of tar 26 J4 gallons. 

Barrel of anchovies 30 pounds. 

Barrel of butter 224 „ 

Barrel of candles 120 „ 

Barrel of turpentine 2 to 2ji c^vt. 

"Barrel of gunpowder 100 pounds. 

Last of gunpowder 24 barrels. 

Measures for Ships, 

I ton, displacement of a ship, 35 cubic feet 

I ton, registered internal capacity of do., 1 00 do. 

I ton, shipbuilders' old measurement, 94 do. 

Comparison of Compound Units. — ^Tables No. 28. 

Measures of Velocity, 

^-y ^^ji. S ^•467 feet per second. 

I mile per hour { 88.0 feet per minute. 

I knot per hour i,688 feet per second. 

I foot per second .682 mile per hour. 

I foot per minute .01136 mile per hour. 



GREAT BRITAIN AND IRELAND. — COMPOUND UNITS. 14$ 

Measura of Volume and Time, 

I cubic foot per second [ ^""^ ^"^!^ ^^^^ P^^' T'"'*^- 

I ^33-333 cubic yards per hour. 

I cubic foot per minute 2.222 cubic yards per. hour. 

I cubic yard per hour .45 cubic foot per minute. 

I cubic inch per second [ ''•^^ cubic foot per hour. 

^ ( 12.984 gallons per hour. 

I gaUon per second 569. 124 cubic feet per hour. 

I gallon per min ute 9-485 cubic feet per .hour. 

Measures of Pressure and Weight (See also page 127.) 

i 144 lbs. per square foot. 

I lb. per square inch < 1296 lbs. per square yard. 

i -57^6 ton per square yar^. 

1 atmosphere (14.7 lbs.) per ) g ^^^ ^^ 

square mch j ^ ^ v ^ j 

i .00694 lb. per square inch. 

I lb. per square foot < .11 ii ounce per square inch. 

( .0804 cwt per square yard. 

,, . , ( 2.0355 inches of mercury at 32° F. 

1 lb. per square inch -^ ^.j^^^f^et of water at s l\z F. 

. , X. 1. o -c f .401 lb. per square inch. 

I inch of mercury at 32 F. -j ^ ^^^ feet of water at S2".3 F. 

( -4333 lb. per square inch. 

I foot of water, at S2°.3 F. .. <| 62.4 lbs. per square foot. 

( .8823 inch of mercury at 32® F. 

Measures of Weight and Volume, 

{405. 1 grains per cubic inch. 
.926 ounce per cubic inch. 
4.107 cwt. per cubic yard. 
1.205 tons per cubic yard. 

, . . , f ^•O'^o ounces per cubic foot. 

1 grain per cubic mch { ^ ^J^ pounds per cubic foot 

I ounce per cubic inch 1 08 pounds per cubic foot. 

1 cwt. per cubic yard 4. 1 48 pounds per cubic foot. 

I ton per cubic yard 8 2. 963 pounds per cubic foot 

r I pound for 1 1 2 2 cubic feet. 



I grain per gallon (i in 70,000 
parts by weight, of water) 



I pound for 41.5 cubic yards. 
I pound for 31.8 cubic metres. 
220 grains for i cubic metre. 
.503 ounce for i cubic metre. 

Measures of Power, 

w. ce \ xj T> ( 1,980,000 foot-pounds per lb. of fuel. 
1 lb. of tuei per n.i'. \ ^21.76 million foot-pounds per cwt of fuel 

P^^ ^^"^ i 2,565 units of heat 

i,ooo,ooofoo^p^^^^ I j^3 p^^^^g ^^f^^i p^^ H.P. per hour. 



146 WEIGHTS AND MEASURES. 

FRANCE.— THE METRIC STANDARDS OF WEIGHTS 

AND MEASURES. 

The primary metric standards are : — the metre, the unit of length ; and 
the kilogramme, the unit of weight, derived from the metre : being the two 
platinum standards deposited at the Palais des Archives at Paris. 

The standard metre is defined to be equal to one ten-millionth part of 
the quadrant of the terrestrial meridian, that is to say, the distance from 
the equator to the pole, passing through Paris, which, by the latest and 
most authoritative measurement, is 39.3762 inches, in terms of the Imperial 
standard at 62° F. By the latest and most accurate measurement, the 
actual standard metre at 32° F. is, in terms of the Imperial standard at 62° F., 
39.37043 inches; and its legal equivalent, declared in the Metric Act of 
1864, is ^9.3708 inches, being the same as that adopted in France. 

The standard kilogramme (looo grammes) is defined to be the weight of 
a cubic decimetre of distilled water at its maximum density, at 4^0 C. 
or 39°. I F. This is legally taken to be 

.2.20462125 lbs., or 
, 2 lbs., 3 oz., 4.383 drachms, or 

i5»432.34874 grains. 

There is in the Standard Department at Westminster a newly-constructed 
subdivided standard yard, laid down upon a bar of Baily's metal, upon 
which a subdivided metre has also been laid down. 

The metric unit of capacity is the litre, defined to be equal to a cubic 
decimetre. Its Imperial equivalent is 0.22009 gallon. 

There is no other official standard of weight and measure in France 
than the metre and the kilogramme; there is no standard litre or unit of 
capacity. 

The metric system is not really founded on the length of a quadrant of the 
meridian, and although it is described as a scientific system, because of the 
simple and definite relation between the metre, which is its basis and unit of 
length, and the kilogramme and litre, which are the units of weight and 
capacity, it is admitted that it has been found impossible practically to 
carry it out with scientific accuracy. The standard kilogramme is admitted 
not to be actually the weight of a cubic decimetre of pure water at the 
specified temperature, nor the litre a measure of capacity holding a cubic 
decimetre of pure water. The real standard unit of weight is declared, even 
by men of science in France, to be merely the platinum kilogramme-weight 
deposited at the Palais des Archives, as the real standard unit and basis of 
the metric system is the platinum metre, also deposited there. It is an 
accomplished fact, however, that all civilized nations have tacitly agreed to 
recognize the metric system as affording for the future the advantages of a 
universal system of weights and measures, and to adopt the standards 
deposited at the Palais des Archives as the primary units of the system. 

The French metric system has been adopted, and its use made compul- 
sory by the following States: — France and Belgium, in 1801; Holland, in 
1819; Greece, in 1836; Italy and Spain, in 1859; Portugal, in 1860--68; 
the German Empire, in 1872; Colombia, Venezuela, in 1872; Ecuador, 



FRANCE.— THE METRIC STANDARDS — LENGTH. I47 

Bcazfl, Peru, and Chili, in i860; also by the Argentine Confederation, and 
Uruguay. 

Great Britain and Ireland, in 1864, adopted the metric system, so far as to 
render contracts in terms of the French metric S3rstem permissive. 

The United States of North America, in 1866, legalized the French metric 
system concurrently with the old system; it was also legalized in British 
North America. 

Switzerland, in 1856, legalized the foot of three decimetres as the unit of 
length, with a decimal scale; the unit of weight being the pound of 500 
grammes, or half a kilogramme, with two distinct scales of multiples and 
parts, one decimal, the other according to the old custom. 

Sweden, in 1855, by a law made compulsory in 1858, adopted a decimal 
system of weights and measures, having for the unit of length a foot of 0.297 
metre, and the unit of weight a pound of 0.42 kilogramme: — ^being the 
original units decimally treated. 

Denmark adopted the metric system so far as the pound of 500 grammes. 
The pound is decimally treated, and since 1863 the use of the greatest 
parts of the multiples of the pound not conformable to decimal sub- 
division has been prohibited. 

Austria, in 1853, adopted a pound of 500 grammes, with decimal divisions, 
for customs and ^scaJ purposes. 

Russia awaits the example of those countries with which she has 
conunercial relations, especially of England. 

In Morocco and Tunis, the weights and measures have no relation with 
the metric system. 

On the 20th May, 1875, the international convention for the adoption 
of the French metrical system of weights and measures was signed at Paris 
by the plenipotentiaries of France, Austria, Germany, Italy, Russia, Spain, 
Portugal, Turkey, Switzerland, Belgium, Sweden, Denmark, the United 
States, the Argentine Republic, Peru, and Brazil. A special clause 
reserves to States not included in the above list the right of eventually 
adhering to the convention. 

I. French Measures of Length. — Table No. 29. 

I millimetre 10 centimetres. 

10 centimetres i decimetre. 

10 decimetres, or ] 

100 centimetres, or > i metre. 

1000 millimetres j 

10 metres i decametre. 

10 decametres x i hectometre. 

10 hectometres, or 1000 metres i kilometre {kilo,) 

10 kilometres i myriametre. 



I toise (old measure) =1.949 metres. 

1000 toises I mille = 1.949 kilometres. 

2000 toises I itinerary league =3.898 „ 

2280.329 toises I terrestrial league =4.444 „ 

2850.411 toises I nautical league =5-555 n 

X noeud (British nautical mile) = 1.855 „ 



148 



WEIGHTS AND MEASURES. 



French Wire-Gauges {Jauges de Fils de Fer), 

The French wire-gauge, like the English, has been subject to variation. 
Table No. 30 contains the values of the "points," or numbers, of the 
Limoges gauge; table No. 31 gives the values of a wire-gauge used in the 
manufacture of galvanized iron; and table No. 32 the values of a gauge 
which comprises wire and bars up to a decimetre in diameter. 





French Wire-Gauge i 


[J<^uge de Limoges) 


.—Table No. 30. 


Number. 


Diameter. 


Number. 


Diameter. • 


Number. 


Diameter. 




MUUmetreJ 


Inch. 




Millimetre. 


Inch- 




Millimetre. 


Inch. 





•39 


.0154 


9 


1.35 


.0532 


18 


3- 40 


.134 


I 


.45 


.0177 


10 


1.46 


•0575 


19 


3-95 


.156 


2 


.56 


.0221 


II 


1.68 


.0661 


20 


4-50 


.177 


3 


.67 


.0264 


12 


1.80 


.0706 


21 


510 


.201 


4 


.79 


.0311 


13 


1.91 


.0752 


22 


5-65 


.222 


5' 


.90 


•0354 


14 


2.02 


•0795 


23 


6.20 


.244 


6 


1. 01 


.0398 


15 


2.14 


.0843 


24 


6.80 


.268 


7 


1. 12 


.0441 


16 


2.25 


.0886 








8 


1.24 


.0488 

1 


17 


2.84 


.112 









French Wire-Gauge for Galvanized Iron Wire. — ^Table No. 31. 



Number. 


Diameter. 


Number. 


Diameter. 


Number. 


Diameter. 




M'metre. 


Inch. 




M'metre. 


Inch. 




M'metre. 


Inch. 


I 


.6 


.0236 


9 


1.4 


.0551 


17 


30 


.118 


2 


.7 


.0276 


10 


1-5 


.0591 


18 


3-4 


.134 


3 


.8 


•0315 


II 


1.6 


.0630 


19 


3.9 


.154 


4 


•9 


•0354 


12 


1.8 


.0709 


20 


4.4 


.173 


5 


I.O 


•0394 


13 


2.0 


.0787 


21 


4.9 


.193 


6 


I.I 


.0433 


14 


2.2 


.0866 


22 


5-4 


.213 


7 


1.2 


.0473 


15 


2.4 


.0945 


23 


5-9 


.232 


8 


1.3 


.0512 


16 


2.7 


.106 









French Wire-Gauge. — Table No. 32. 



Mark. 


Size. 


Mark. 


Size. 


Mark. 


Size. 


Mark. 


Size. 




Millimetre. 




Millimetre. 




Millimetre. 




Millimetre. 


P 


5 


8 


13 


16 


27 


24 


64 


I 


6 


9 


14 


17 


30 


25 


70 


2 


7 


10 


IS 


18 


34 . 


26 


76 


3 


8 


II 


16 


19 


39 


27 


83 


4 


9 


12 


18 


20 


44 


28 


88 


5 


10 


13 


20 


21 


49 


29 


94 


6 


II 


14 


22 


22 


54 


30 


100 


7 


12 


15 


24 


23 


59 







FRANCE. — THE METRIC STANDARDS. I49 

II. French Measures of Surface. — Table No. 33. 

1 00 square millimetres i square centimetre. 

1 00 square centimetres i square decimetre. 

100 square decimetres, or ) 

^ ^. ^ ' > I square metre, or centiare. 

r 0,000 square centimetres / ^ ^ '-^ > ^ ^ 

100 square metres, or centiares... i square decametre, or are. 
100 square decametres, or ares ... i square hectometre, or hectare. 
100 square hectometres, or hectares 1 square myriametre. 

liand is measured in terms of the centiare^ the are, and the hectare or 
arpent metrique {metric acre). There is also the decare, of 10 ares. 

III. French Measures of Volume. — Tables No. 34. 

Cubic Measure. 

1 000 cubic millimetres i cubic decimetre. 

1000 cubic decimetres i cubic metre. 

Wood Measure, 

10 decist^res i stfere* (i cubic metre). 

I voie (Paris) 2 stferes. 

I voie de charbon (charcoal) 0.2 stfere ( 75 cubic metre). 

I corde 4 stferes. 

* The stire measures 1. 14 metres x 0.88 metre x i metre, the billets of wood being 

1. 14 metre in length. 

IV. French Measures of Capacity. — ^Tables No. 35. 

Liquid Measure, 

10 centilitres i decilitre. 

10 d^cihtres i litre. 

10 litres I decalitre. 

Dry Measure. 

10 litres I decalitre. 

10 decalitres, or) ^ hectoUtre. 

100 litres J 

10^ liS"*'^' °'} ' '^""^^ <' ^^'*= ™*''^>- 

The use of measures equal to a double-litre^ a half-litre, a double-dicilitre^ a 
half-d£cilitre^ is sanctioned by law. 



ISO 



WEIGHTS AND MEASURES. 



V. French Measures of Weight. — Table No. 36. 

10 milligrammes i centigramme. 

10 centigrammes i decigramme. 

10 decigrammes i gramme. 

10 grammes i decagramme. 

10 decagrammes i hectogramme. 

10 hectogrammes, or ) ^ kilogramme (M., kUc^.) 

1000 grammes J \ > c / 

10 kilogrammes i myriagramme. 

10 myriagtammes, or ) ^ j^ ^ ^^ 

100 kilogrammes J ^ ^ 

10 quintaux, or ) ( i millier, tonneau de mer, or tonne 

1 000 kilogrammes j ( (weight of i cubic metre of water at 39°. i ). 



EQUIVALENTS OF BRITISH IMPERIAL AND FRENCH METRIC 

WEIGHTS AND MEASURES. 



I. Measures of Length. — ^Tables No. 37. 



A DBCIMBTRB DIVIDBD INTO CBNTIMBTRBS AND MILLIMBTRBS. 



2 



a 



llljl.nlllllllHIllllliJhllllini.ll.limlhTT 



llllllill 



JTTTT 



'mill. I 



'"iImii 



iiimm 



UL 



It I I I I I I I I 11 M |l 



IIMII erg: 



MM 



' ' ' ' 



I I I .1 i! 



INCHBS AND TBNTHS. 



Mbtsic Dbnominations 

AND VaLUBS. 


• 

Equivalbnts in Impbrial Dbnominations. 




Metres. 


Inches. 


Feet. 


Yards. 

« 


MUes. 


I millimetre 
I centimetre 
I decimetre 

I M£1K£ «•.. 

I dekametre 
I hectometre 

I KILOMETRE 

I myriametre 


/xooo 

/lOO 

V.O 

I 

10 

100 

1,000 
10,000 


= 0-03937 
= 0.39370 

= 3-93704 
= 39-37043 


= 3.28087 
= 32.80869 

= 3280.87 


= 1.09362 
10.93623 
109.36231 
= 1,093.6231 
= 10,936.231 


= 0.62138 
= 6.21377 



IMPERIAL AND METRIC EQUIVALENTS. 



IS! 



Tables No. 37 {continued). 



Impbrial Denominations. 


Equivalents in Metric Denominations. 


Centunetres. 


Metres. 


Kilometres. 


1 

I inch I2K.A millimetres) 


= 2.5399s 


= , 0.30480 

0.91439 

1.82878 

5.02915 

= 20.11662 

= 201.1662 

= 1,609.3296 


= 0.20117 
= 1.60933 


I foot, or 12 inches 

I yard, or 3 feet, or 36 inches.... 
I fathom, or 2 yards, or 6 feet.... 
I pole, OT K'^A yards 


I chain, or 4 poles, or 22 yards... 
I furlong, 40 poles, or 220 yards 
I mile, 8 furlongs, or 1760 yards 



Equivalent Values of Millimetres and Inches. — ^Tables No. 38. 



Millimetres = Inches. 



MiOimetRs. 


Inches. 


MUiimetres. 


Inches. 


Millimetres. 


Inches. 


Millimetres. 


Inches. 


I 


.0394 


27 


1.0630 


53 


2.0866 


79 


3-"03 


2 


.0787 


28 


1. 1024 


54 


2.1260 


80 


3.1496 


3 


.1181 


29 


I.I417 


55 


2.1654 


81 


3.1890 


, 4 


-1575 


3<5 


I.181I 


56 


2.2047 


82 


3.2284 


5 


.1968 


31 


1.2205 


57 


2.2441 


83 


3.2677 


6 


.2362 


32 


1.2598 


58 


2.2835 


84 


3-3071 


7 


.2756 


33 


1.2992 


59 


2.3228 


85 


3-3465 


8 


.3150 


34 


1.3386 


60 


2.3622 


86 


3-3859 


9 


•3543 


35 


1.3780 


61 


2.4016 


87 


3-4252 


10 


.3937 


36 


1-4173 


62 


2.4410 


88 


3.4646 


II 


.4331 


37 


1.4567 


63 


2.4803 


89 


3-5040 


12 


.4724 


38 


1.496 1 


64 


2.5197 


90 


3-5433 


13 


.5118 


39 


1-5354 


65 


2.5591 


91 


3-5827 


14 


.5512 


40 


1.5748 


66 


2.5984 


92 


3.6221 


; '5 


.5906 


41 


1. 6142 


67 


2.6378 


93 


3.6614 


16 


.6299 


42 


1.6536 


68 


2.6772 


94 


3.7008 


17 


.6693 


43 


1.6929 


69 


2.7166 


95 


3-7402 


18 


.7087 


44 


1-7323 


70 


2.7559 


96 


3.7796 


19 


.7480 


45 


1.7717 


71 


2.7953 


97 


3.8189 


20 


.7874 


46 


1.8110 


72 


2.8347 


98 


3.8583 


1 21 


.8268 


47 


1.8504 


73 


2.8740 


99 


3-8977 


'- 22 


,8661 


48 


1.8898 


•74 


2.9134 


100 


3-9370 


23 


.9055 


49 


1. 9291 


75 


2.9528 


= 1 de 


cimetre. 


24 


.9449 


50 


1.9685 


76 


2.9922 






' 25 


.9S43 


51 


2.0079 


77 


3-0315 






i ^' 


1.0236 


52 


2.0473 


78 


3.0709 







152 



WEIGHTS AND MEASURES. 



Tables No. 38 {continued), 

InCHBS DbCIMALLY = MiLLIMBTRBS. 



Inches. 


1 
Millimetres, i 


Inches. 


Millimetres. 


Inches. 


Millimetres. 


Inches. 


Millimetres. 
239 


.01 


-5 1 


.26 


6.60 


.60 


152 


.94 


.02 


•51 


.28 


7.II 


.62 


15-7 


.96 


24.4 


•03 


.76 


•30 


7.62 


.64 


16.3 


.98 


24-9 


.04 


1.02 


.32 


8.13 


.66 


16.8 


1. 00 


25.4 


•05 


1.27 


•34 


8.64 


.68 


17-3 


2.00 


50.8 


.06 


1-52 


.36 


9.14 


.70 


17.8 


3.00 


76.2 


.07 


1.78 


.38 


9^65 


.72 


18.3 


4.00 


IOI.6 


.08 


2.03 


.40 


10.2 


.74 


18.8 


5.00 


127.0 


.09 


2.29 


.42 


10.7 


.76 


19-3 


6.00 


152.4 


.10 


2.54 


.44 


II. 2 


.78 


19.8 


7.00 


177.8 


.12 


3.05 


.46 


II.7 


.80 


20.3 


8.00 


203.2 


.14 


3.56 


.48 


12.2 


.82 


20.8 


9.00 


228.6 


.16 


4.06 


•50 


12.7 


.84 


21.3 


10.00 


254.0 


.18 


4.57 


•52 


13.2 


.86 


21.8 


11.00 


279.4 


.20 


5.08 


•54 


13-7 


.88 


22.4 


12.00 


304.8 


.22 


5-59 


.56 


14.2 


.90 


22.9 


= I 


foot. 


.24 


6.10 


.58 


14.7 


•92 


23.4 











Inches 


IN Fractions = Millimetrks. 






Eighths. 


Sixteenths. 


Thirty-seconds. 


Millimetres. 


Eighths. 


Sixteenths. 


Thirty-seconds. 


Millimetres. 






I 


•79 






17 


135 




I 


2 


1-59 




9 


18 


143 






3 


2.38 






19 


I5-I 


I 


2 


4 


3-17 


5 


10 


20 


159 






5 


3-97 






21 


16.7 




3 


6 


4.76 




II 


22 


175 






7 


5-56 






23 


18.3 


2 


4 


8 


6.35 


6 


12 


24 


19.0 






9' 


7.14 






25 


19.8 




5 


10 


7^94 




13 


26 


20.6 






II 


8.73 






27 


21.4 


3 


6 


12 


9^52 


7 


14 


28 


22.2 






13 


10.32 






29 


23.0 




7 


14 


II. II 




15 


30 


23.8 






15 


II. 91 






31 


24.6 


4 


8 


16 


12.7 8 


16 


32 


25.4 



By means of the preceding tables of equivalent values of inches and 
millimetres, the equivalent values of inches in centimetres and decimetres, 
and even in metres, may be found by simply altering the position of the 
decimal point. This method naturally follows from the decimal subdivisions 
of French measure. 

Take, for example, the tabular value of i millimetre, and shift the 



IMPERIAL AND METRIC EQUIVALENTS. 



153 






decimal pK>int successively, by one digit, towards the right-hand side; the 
values of a centimetre, a decimetre, and a metre are thereby expressed in 
inches, as follows: — 

I millimetre 0394 inches. 

I centimetre o-394 

I decimetre 3.94 

I metre 39.4 

At the same time, it appears that, by selecting the tabular value of 
10 millimetres, the value of its multiples are given more accurately, thus, — 

10 millimetres, or i centimetre 0-3937 inches. 

I decimetre 3-937 »» 

I metre 39.37 „ 

Again: — 

100 millimetres, or i decimeti^ = 3-937 inches. 

I metre =39-37 n 

Similarly, for example : — 

.32 inch = 8.13 millimetres. 
3.2 „ = 81.3 „ 

^ f 813.0 „ or 

( .813 metre. 



32.0 



» 



II. Square Measures, or Measures of Surface. — Tables No. 39. 



Mbtric 



I square centimetre 

1 square decimetre 

I square metre, or centiare 

I ARE, or square dekametre, or 100 
square metres 

1 hectare, or metrical acre, or 100 ares, 
or 10,000 square metres 






Imperial Square Measures. 

.155 square inch. 

15.5003 square inches. 

10.7641 square feet, or 

1. 1960 square yards. 

1076.41 square feet, or 

119.60 square yards. 
1,960.11 square yards, or 

2.4711 acres, or 
acres and 2280.1240 square 
yards. 



Imperial = Metric Square Measures. 



I 

Imperial Measures. 


S<}uare 
Centimetres. 


Square Metres. 


Ares. 


Hectares. 


f snuare incli r 


= 6.45148 


= 0.092901 
= 0.836 II 2 

= 25.292 

= 1011.696 

= 4046.782 


= 10.11696 
= 40.4678 


= 0.40468 
= 258.98944 


1 square ft., or 144 sq. inches 

I square yard, or 9 square ) 

tcet, or 1296 sq. inches ) 

I perch or rod, or 30X | 

square yards y 

I rood, or 40 perches, or") 

I 1 2 10 square yards ) 

I acre, or 4 roods, or 4840 ) 

square yards J 

I square mile, or 640 acres 



154 



WEIGHTS AND MEASURES. 



III. Cubic Measures. — Tables No. 40. 

Metric = Imperial Cubic Measures. 

I cubic centimetre = 0.061025 cubic inch. 

u* J -^ 4- (61.02524 cubic inches, or 

I cubic deametre = | 0.0353156 cubic foot 

, . _, , f 35.^156 cubic feet, or 

I cubic metre = { 1.308 cubic yanis. 

Imperial = Metric Cubic Measures. 

1 cubic inch = 16.387 cubic centimetres. 

I cubic foot 



I cubic yard 



_ 128.31531 cubic decimetres, or 
~ ( 0.02I 



1283 1 6 cubic metre. 
0.76453 cubic metre. 



Wood Measure. 



 «'«. " »"« »"" { ^fisfcS^jSa 

I decistfere * 3.5316 cubic feet. 

I voie de bois (wood), or 2 stores. Paris { '^it'.l^^^y^^l 

I voie de charbon (charcoal) = i sack ( S}^ bushels, or 

= */5 stfere ( 7.063 cubic feet 

I corde of wood = 4 cubic metres 141.26 cubic feet 

IV. Measures of Capacity. — Tables No. 41. 



or 



Metric Dekominations 
AND Values. 


Equivalents in Imperial Denominations. 




Litres. 


Gills. 


Pints. 


Quarts. 


Gallons. 


Bushels. 

a 


Quarters. 


Centilitre 

Decilitre 

Litre 
(61.02524c. in.) 

Dekalitre 

Hectolitre 

Kilolitre 


/lOO 

}' 

10 

100 

1000 


0.0704 
0. 7043 


0.0176 
O.I761 

1.7607 


0.8804 


0.2201 

2.2009 
22.009 
220.09 


0.2751 
2.75II 
27.511 


0.344 
3.439 



Imperial Denominations. 


Equivalents in Metric Denominations. 


Litres. 


Dekalitres. 


Hectolitres. 


I gill 

I pint, or 4 gills 

I quart, or 2 pints 

I gallon, or 4 Quarts 


= 0.1420 
= 0.5679 

= I.I359 

= 4.5436 
= 9.0872 

= 36.3488 
= 290.7904 


= 0.9087 

= 3.6349 
= 29.0790 


= 2.9079 


I peck, or 2 crallons 


I bushel, or 8 gallons 

I Quarter, or 8 bushels 





IMPERIAL AND METRIC EQUIVALENTS. 



ISS 



V. Measures of Weight. — Tables No. 42. 



MsTKic Weights = Impbkial AvoiitDUPOis Weights. 

I kilogramme = 2 lbs. 3 oz. 4 drachms^ 10.47374 grains. 



Mbthic Weights. 


Equivalbnts im Imperial Denominations. 




Grammes. 


Grains. 


Ounces. 
1 


Pounds. 


Hundred- 
weights. 


Tons. 


M illigramme 


Vxooo 
Vioo 

I 

10 

100 

1,000 

10,000 

100,000 

1,000,000 


0.01 54 
0.1543 

1.5432 

154323 

154.3235 

1543.2349 
15432.3487 


0.3527 

3.5274 

35.2739 


2.2046 

22.0462 

220.4621 

2204.6212 


1.9684 
19.6841 


1 II ! lit 


CentigrazDzne 

Decieramme 


Gramme 

Dekagramme 

Hectogramme 

Kilogramme 

Myriagramme 

Quintal^ or 100 kilog. 
Millier, or metric ton 



Imperial Avoirdupois = Metric Weights. 



iMPBfiiAi. Avoirdupois 
Weights. 


Grammes. 


Decigrammes. 


Kilogrammes. 


Millier, or 
Metric Ton. 


I diachni 


= I.77184 
- 28.34954 
= 453.59265 


= 2.83495 
= 45.35926 


0.45359 
== 50.80237 

= IO16.O4754 


= 1. 01604 


I ounce, or 16 drams 
I pound, or 1 6 ounces 
I hundredweight, ) 

or 112 pounds ) 
I ton, or 20 hun- \ 

dredweights j 



Metric Weights = Imperial Troy Weights. 

I kilogramme = 2 troy lbs. 8 oz. 3 dwts., .34874 grain. 



Hetkxc Weights. 


Grains. 


Pennyweights. 


Ounces. 


Troy Pound. 


1 

Milligramme... 


0.01543 




^-i 




Centigramme ... 


0.15432 








Deagramnie . . . 


1.54323 








GRABHtfE. 


15-43234 








Dekagramme... 


= '154.32349 


= 0.64301 


= 0.32151 




Hectogramme.. 


= 1543.23487 


= 6.43014 


= 3-21507 




Kilogramme... 


= I51432.34874 




= 32.15073 


= 2.67922 



156 



WEIGHTS AND MEASURES. 



Imprxial Troy = Metric Weights. 



Imperial Troy Weights. 


Equivalents in Metric Denominations. 


Millignuiime. 


Gramme. 


Dekagramme. 


Hecto- 
gramme. 


Kilo- 
gramme. 


I troy grain 

I „ dwt, or 24 gr. 
I „ oz., or 480 „ 
I „ lb., or 5,760 „ 


64.79895 


0.06480 

1-55517 

31.10349 

373-24195 


3.IIO35 
37.32419 


373242 


0-37324 



APPROXIMATE EQUIVALENTS OP ENGLISH AND 

FRENCH MEASURES. 

The following are approximately equal English and French measures of 
length : — 

I pole, or perch {s}4 yards)... 5 metres (exactly 5.029 metres). 

I chain (22 yards) 20 metres (exactly 20. 1 1 66 metres). 

I furlong (220 yards) 200 metres (exactly 201.166 metres). 

5 furlongs I kilometre (exactly i .0058 kilometres). 

^ . (3 decimetres (exactly 3.048 decimetres), or 

^ ^ \ 30 centimetres. 

One metre = 3.28 feet = 3 feet 3 inches and 3 eighths all but Vsxa inch; 
= 40 inches nearly ( ^/e^th. or 1.6 per cent less). 

.100 metre (i decimetre) 

.010 metre (i centimetre) , , __ , , ^. 

.001 metre (i millimetre) = .04 inch, or Viooths inch, or two-thirds 

of Vx6 inch, or 725 inch, nearly. 

One inch is about 2}4 centimetres (exactly 2.54). 

One inch is about 25 millimetres (exactly 25.4). 

One yard is "/"ths of a metre. 11 metres are equal to 12 yards. 

Approximate rule for converting metres, or parts of metres, into yards : — 
Add Vxi^h {}( per cent. less). 

For converting metres into inches: — Multiply by 40; and to convert 
inches into metres, or parts of metres, divide by 40. 

One kilometre is about ^ mile (it is 0.6 per cent. less). 

One mile is about 1.6 or 1 3/^ kilometres (it is 0.6 per cent less) == 16 10 
metres, about. 

« 

With respect to superficial measures : — 
One square centimetre is about 7*6.5 part of a square inch. 
One square inch is equal to about 6.5 square centimetres. 
One square metrecontains fully 10 J^ square feet, or nearly i^j square 3rards. 
One square yard is nearly ^7 ths of a square metre. 
One acre is over 4000 square metres (about 1.2 per cent more). 
One square mile is nearly 260 hectares (about 0.4 per cent less). 



FRENCH AND ENGLISH COMPOUND UNITS. 1 57 

With respect to cubic measures, and to capacity : — 

One cubic yard is about 5^ cubic metre (it is 2 per cent. more). 
One cubic metre is nearly i^ cubic yard (it is i^ per cent. less). 
One cubic metre is nearly 35 'A cubic feet (it is .05 per cent. less). 
One litre is over i^ pints (it is 0.57 per cent more). 
One gallon contains above 4^ litres (it holds about i per cent. more). 
One kilolitre (a cubic metre) holds nearly i ton of water at 62*^ F. (i^ 
per cent less). — One cubic foot contams 28.3 litres. 

With respect to weights: — The ton and the gramme stand at nearly 
equal distances above and below the kilogramme, thus : — 

I ton is 1,016,047.5 grammes, 

I kilogramme is 1,000.0 grammes, 

I gramme i.o gramme, 

in the ratio of about 1,000,000 : 1,000 : i. 

One gramme is nearly 15^ grains (about yi per cent. less). 

One kilogramme is about 2 '/s pounds avoirdupois (about V4 per cent, 
more). 

A thousand kilogrammes, or a metric ton, is nearly one English ton 
(about I J^ per cent less). 

One hundredweight is nearly 5 1 kilogrammes ( 2/5 per cent less). 



EQUIVALENTS OF FRENCH AND ENGLISH COMPOUND 

UNITS OF MEASUREMENT. 

Weighty Pressure^ and Measure, 

• kiX*"-- P« "■""••■ { ;IM JSjl'^Taii 

I pound per foot 1.488 kilogrammes per metre. 

1 pound i>er yard .496 kilogramme per metre. 

1000 kilc^ammes per metre .300 ton per foot 

I ton per foot 3333-333 kilogrammes per metre. 

1000 kfl^mmes, or X tonne, per ) ^^g^ ^^^ ^^ ^^^ 

I tcm per mile 631.0 kilogrammes per kilometre. 

, T .„../ 1422.32 pounds per square inch. 

I kilogramme per square millimetre \ l^^\^^^ ^^^ ^^^^^^^ ^^^^ 

1000 Tiounds ner souare inch i -703077 kilogramme per square 

1000 pounas per square men | millimetre. 

, ton per square inch { '"575 ^"^JSTeSe.^" ^' . 

I kilogramme per square centimetre 14.2232 pounds per square inch. 

1-0335 kilogrammes per square centi- ) nounds oer souare inch 

metre (i atmosphere) / ^^'^ pounds per square men. 

I pound per square inch 



I pound per square foot 



.0703077 kilogramme per square 
centimetre. 
4.883 kilogrammes per square 
metre. 



158 



WEIGHTS AND MEASURES. 



Weighty Pressure^ and Measure {continued), 

kilogramme per square metre 205 pounds per square foot. 

centimetre of mercury 394 inch of mercury. 

inch of mercury 2. 540 centimetres of mercury. 

centimetre of mercury 193 pound per square inch. 

pound per square inch 5-i7o centimetres of -mercury. 

gramme per litre 70.105 grains per gallon. 

grain per gallon 0143 gramme per litre. 

kilogramme per cubic metre 0624 pound per cubic foot 

pound per cubic foot 16.020 kilogrammes per cubic metre. 

. ^ u- _ * f .984 ton per cubic metre. 

tonne per cubic metre < \. *, u- ^ 

^ ( .752 ton per cubic yard. 

kilogramme per litre 10.016 pounds per gallon. 

pound per gallon 998 kilogramme per litre. 

ton per cubic metre 1.016 tonnes per cubic metre. 

ton per cubic yard i'329 tonnes per cubic metre. 

cubic metre per kilogramme 16.020 cubic feet per pound. 

cubic foot per pound 06 24 cubic metre per kilogramme. 

( 1.329 cubic yards per ton. 
cubic metre per tonne < 1.794 cubic feet per cwt. 

I 35.882 cubic feet per ton. 

cubic yard per ton 752 cubic metre per tonne. 

cubic foot per cwt. . . .^ 557 cubic metre per tonne. 

cubic foot per ton 0279 cubic metre per tonne. 

Volumty Area, and Length, 

cubic metre per lineal metre 1.196 cubic yards per lineal yard. 

cubic yard per lineal yard ..• 836 cubic metre per lineal metre. 

cubic metre per square metre 3.281 cubic feet per square foot 

cubic foot per square foot 3.048 cubic metres per square metre- 
litre per square metre 0204 gallon per square foot 

gallon per square foot 48. 905 litres per square metre. 

i .405 cubic metre per acre, 
cubic metre per hectare \ .529 cubic yard per acre. 

( 89.073 gallons per acre. 

cubic metre per a^re 2.471 cubic metres per hectare. 

cubic yard per acre 1.902 cubic metres per hectare. 

000 gallons per acre 11.226 cubic metres per hectare. 

Work, 

kilogrammetre {ky.m) 7.233 foot-pounds. 

foot-pound 138 kilogrammetre. 

cheval-vapeur or cheval (75 -t x « ) horse-power. 

persecond) / -^ ^ ^ 

horse-power 1.0139 chevaux. 

kilogramme per cheval 2.235 pounds per horse-power. 

pound per horse-power 447 kilogramme per cheval. 

square metre per cheval 10.913 square feet per horse-power. 

square foot per horse-power 0916 square metre per cheval. 

cubic metre per cheval • 35.801 cubic feet per horse-power. 

cubic foot per horse-power 0279 cubic metre per cheval. 



FRENCH AND ENGLISH COMPOUND UNITS. 



159 



Heat. 



I calorie, or French unit 
I English heat-unit 



French mechanical equivalent (424 ) , 
kilogrammetresj exactly 423.55) ... j "^^ ^' 

English mechanical equivalent (772 ) 
foot-pounds) J 

I calorie per square metre 

I heat-unit per square foot 

I calorie per kilogramme 

1 heat-unit per pound 



10. 

• 

2. 
I. 



968 English heat-units. 
252 calorie. 

5 foot-pounds. 

76 kilogrammetres. 

369 heat-unit per square foot. 
713 calories per square metre. 
800 heat-units per pound. 
556 calorie per kilogramme. 



Speedy 6fc. 



I 



I metre per second 

I kilometre per hour 

I foot per second, or per minute < 

^ \ 1.609 kilometres per hour. 

, ^.t- _^. J f 35.316 cubic feet per second. 

I cubic metre per second < ^^^ ^ x^- r ^ • *^ 

* ( 2119 cubic feet per mmute. 

I cubic foot per second, or per minute 

I cubic metre per minute 

I cubic yard per minute 



3.281 feet per second. 
196.860 feet per minute. 
2.236 miles per hour. 
.621 mile per hour. 
.305 metre per second, 

minute. 
.447 metre per second. 
1.609 kilometres per hoi 



or per 



.02 



Money, 



I firanc per kilogramme 



I penny per pound 

I shilling per pound 

I shilling per cwt., or £,1 per ton... < 

I franc per quintal 

I franc per tonne < 

I fianc per metre \ 

1 shilling per yard 

I franc per kilometre < 

£1 per mile 

I penny per mile 

I franc per square metre \ 



:o cubic teet per secona. 
cubic feet per minute. 
583 cubic metre per second, 
or per minute. 
1.308 cubic yards per minute. 
.765 cubic metre per minute. 



4.320 pence per pound. 
.360 shilling per pound. 
40.320 shillings per cwt., or 
^^40.32 per ton. 
.231 fra^c per kilogramme. 
2.772 franc per kilogramme. 
24.802 francs per tonne. 
2.48 francs per quintal. 
.403 shilling per cwt. 
.484 penny per cwt. 
.806 shilling per ton. 
.726 shilling per yard. 
8.709 pence per yard. 
1.378 francs per metre. 
.0638 J[, per mile. 
15.326 pence per mile. 
15.660 francs per kilometre. 

.0652 francs per kilometre. 
8.028 pence per square yard. 
.669 shilling per square yard- 



l6o WEIGHTS AND MEASURES. 

I shilling per square yard 1.510 francs per square metre. 

jCi per square yard 30. 194 francs per square metre. 

{.270 penny per cubic foot 
7.281 pence per cubic yard 
.607 shilhng per cubic yard. 
•0303 £ per cubic yard. 

I penny per cubic foot 3. 708 francs per cubic metre. 

I penny per cubic yard 137 franc per cubic metre. 

I shilling per cubic yard i .648 francs per cubic metre. 

j£i per cubic yard 32.962 francs per cubic metre. 

I fianc per Utre { 43-270 pence per gallon. 

^ \ 3.606 shillmgs per gallon. 

I franc per hectolitre 1-893 shillings per hogshead (wine). 

I shilling per hogshead 528 franc per hectolitre. 



GERMAN EMPIRE. — ^WEIGHTS AND MEASURES. — Tables No. 43. 

From the ist January, 1872, the French metric system of weights and 
measures became compulsory throughout the German Empire, as follows : — 

I. German Measures of Length. 

French Measure. 

I Strich = I millimetre, 

10 Strichs I New-Zoll = i centimetre. 

100 New-Zolls I Stab = i metre. 

10 Stabs I Kette = i dekametre. 

100 Kettes I Kilometre = i kilometre. 

7 Kilometres i Mile - i 7°°^ metres, or 

7 Kilometres i Mile - | ^^^ ^^^gj-^j^ ^y^^^ 

II. German Measures of Surface. 

I Quadrat-Stab = i square metre. 

1 00 Quadrat-Stabs i Ar = 1 00 square metres. 

. Tj f _ / '^°^ square metres, or 

'°^^^ I nectar ^i .247 acre. 



={'■ 



III. German Measures of Capacity. 

I Schoppen = j4 litre. 

(Beer Measure.) 

2 Schoppens i Kanne = i litre. 

5° Cannes i Scheffel (bushel) = { ^^.Jj'l^^^erial bushels. 

' S^J^effels X Fass (cask) = { J.Jf^^;- 

The kanne is further divided into measures of j^ kanne, ^ kanne, and 
^/x6 kanne. 



GERMAN EMPIRE. — WEIGHTS, THE FUSS. l6l 

IV. German Measures of Weight. 

I Milligramm = i milligramme. 

lo Milligramms i Centigramm = i centigramme. 

lo Centigramms i Dezigramm = i decigramme. 

loo Dezigramms i New-Loth = | '"^ grammes, or 

I -35273 ounce. 
( 500 grammes, or 
50 New-Loths........ i Pfund = < ^ kilogramme, or 

( I . I o 2 3 pounds avoirdupois. 

100 Pfunds I Centner = \ 5° kilogrammes, or 

\ 110.23 pounds avoirdupois. 

2 Centners, or ) Tonne = i ^°^ kilogrammes, or 

200 Pfunds / '" ~ I 220.46 pounds avoirdupois. 



OLD WEIGHTS AND MEASURES OF THE GERMAN STATES. 

These vary for every state. The chief measures of length are the Fuss, 
and the EUe, of which the second is in general twice the first. The 
following are the values of the Fuss, which is the German foot, in the 
principal states. 

Values of the German Fuss in the States and Free Towns of 

THE German Empire. — Table No. 44. 



Prussia 

Bavaria 

Wiirtembeig 

Saxony 

Baden 

Mecklenburg-Schwerin 

Hesse-Darmstadt 

Hesse-Cassel 

Oldenburg 

Brunswick 

Hanover 

Mecklenburg-Strelitz 

Anhalt 

Saxe-Coburg-Gotha 

Saxe-Altenbuig 

Waldeck 

Lippe 

Schwarzburg-Rudolstadt 

Schwarzburg-Sondershausen : — 

(i) High Sovereignty and Amstadt ... 

(2) Low Sovereignty and Sondershausen 

Reuss 

Schaumburg-Lippe 

Hamburg 

Liibeck 

Bremen 



2.356 inches. 

1.491 

1.279 

1.149 

1.811 

I.4S7 

9-843 
1.328 

1.649 

1.235 
1.500 

I.4S7 
2.356 

1.324 
1. 122 

1. 512 

1.398 

5-047 



1. 149 

1.331 

1.280 

1. 421 
1.283 

1.324 

1.392 

11 



i9 



99 



99 



99 



99 



99 



99 



99 



99 



99 



99 



99 



99 



99 



99 



99 



99 



99 



99 



99 



99 



99 



99 



99 



l62 WEIGHTS AND MEASURES. 

KINGDOM OF PRUSSIA.— Old Weights and Measures.— 

Tables No. 45. 

I. Prussiai* Measures of Length. 

English Measure. 

I Linie = .0858 inch. 

12 Linien i ZoU = 1.0297 inches. 

12 ZoU I Fuss = j "-356 inches, or 

I 1.0297 feet. 

2 Fuss I EUe = 2.0596 feet 

1!?!?°'} iRuthe = 4.1192 yards. 

--R"*- xMeae = { ^^^^Ja^^taS. 

Used by Miners, 

I Lachterlinie = .0927 inch. 

I o Lachterlinien i Lach terzoU = .9268 inch. 

10 Lachterzoll i Achtel = .7723 foot 

8 Achtels, or I t i.*. a jx 
5 p^gg I I Lachter = 2.0596 yards. 

9 Fuss i Spanne =6.1788 yards. 

Surveyor^ Measure, 

I Scrupel = .0148 inch. 

10 Scrupel I Linie = .1483 inch. 

I o Linien i Zoll = 1.4828 inches. 

10 ZoU I Land-Fuss = 1.2356 feet. 

10 Land-Fuss i Ruthe =4.1192 yards. 

2000 Ruthen i Meile = 4.6809 miles. 

II. Prussian Measures of Surface. 

I Square Linie = .00736 square inch. 

144 Square Linien i Square Zoll - 1.0603 square inches. 

144 Square Zoll i Square Fuss = 1.0603 square feet 

144 Square Fuss i Square Ruthe =16.967 square yards. 

180 Square Ruthen... i Morgen = .63103 acre. 

30 Morgen i Hufe = 18.931 acres. 

III. Prussian Measures of Volume. 

Cubic Measure, 

I Cubic Linie = .000632 cubic inch. 

1728 Cubic Linien.... i Cubic Zoll = 1.092 cubic inches. 

1728 Cubic Zoll I Cubic Fuss = 1.092 cubic feet 

1728 Cubic Fuss I Cubic Ruthe =69.893 cubic yards. 

For measuring stone and brickwork, earth, peat, fascines, and firewood, 
the following are u§e4 ;— 



PRUSSIA. — CAPACITY, WEIGHTS. 1 63 

I Cubic Klafter, or ) . , . . 

. 108 Cubic Fuss ; - " 7.93 cubic feet 

4ji Klafters i Haufe =530.70 

I Schachnithe (in architecture) 144 Cubic Fuss = 157.25 



99 



IV. Prussian Measures of Capacity. 

Dry Measure, 
I Maasche = •7560 quart. 

JSS'"'"} ^""^^ = 3-4. quarts. 

4 Metzen .. i Viertel = 3.0242 gallons. 

4 Viertel, or ) ^ Scheffel = / ^'S'^' bushels, or 

48 Quarts ) ( i. 941 cubic feet. 

4 Scheffeln i Tonne = 6.0484 bushels. 

if 2S?r } ' Maker = 2..68rs quarters. 

6jSeSn°'} '^' = I X.3407 quartets. 

The Tonne in the table is the measure for salt, lime, and charcoal. 
A Tonne of flax-seed is 2.354 Scheffeln. 

Liquid Measure (for Wine and Spirits). 

32 Cubic ZoU I Ossel = 1.0079 pints. 

2 Ossel I Quart = 1.0079 quarts. 

i:§Slf'" } ^^''^ = 7.559 gallons. 

2 Ankers i Eimer = 15. 118 

2 Elmers i Ohm = 30.237 

ffS°' } ^Oxhoft =45.355 

V. Prussian Measures of Weight. 

I Com = 4. 1 15 grains. 

10 Corns I Cent = .09406 dram. 

10 Cents I Quentche = .9406 dram. 

10 Quentchen i Loth = ,588 ounce. 

30 Loth I Zollpfund = 1. 1023 pounds. 

1 00 Zollpfund I Centner = 110.23 pounds. 

20 Zollpfund I Stein = 22.046 pounds. 






330.69 pounds, or 
2.506 hundredweights. 
4409.2 pounds, or 



3 Centners i Schiffspfund = ^ 

40 Centners I x Schiffslast = { '^'^"^-g^g™; 

The Tonne of coals is 2270 pounds avoirdupois, or 1.013 tons. 



1 64 WEIGHTS AND MEASURES. 

KINGDOM OF BAVARIA.— Old Weights and Measures.— 

Tables No. 46. * 

I. Bavarian Measures of Length. 

 

I Linie = .6798 inch. 

12 Linien i ZoU = .95756 inch. 

12 Zoll I Fuss = .95756 foot 

6 Fuss i Klafter=5.74536 feet. 

10 Fuss i Ruthe =9.5756 feet 

In surveying, the Fuss is divided into 10 Zoll, and i Zoll into 10 Linien. 
The EUe contains 2 Fuss lo^ Zoll, = 2.733 ^^^^ 

II. Bavarian Measures of Surface. 

I Square Zoll = .91692 square inch. 

144 Square Zoll .... i Square Fuss = .91692 square foot 

100 Square Fuss ... i Square Ruthe = 10.188 square yards. 

400 Square Ruthen / ' Tagwerk Morgen, ) ^ ( 4075.188 square yards, or 
^ I or Juchert J | .842 acre. 

III. Bavarian Measures of Volume. 

I Cubic Zoll = .878 cubic inch. 

1728 Cubic Zoll I Cubic Fuss= .878 cubic foot 

126 Cubic Fuss (6x6 xyA Fuss) i Klafter = / "^'•^28 cubic feet or 

' ( 4.097 cubic yards. 

IV. Bavarian Measures of Capacity. 

Dry Measure, 

I Dreisiger= .12745 peck. 

4Dreisigers i Maassl = .12745 bushel. 

4 Maassls i Viertel = .5098 bushel. 

2 Viertel i Metze =1.0196 bushels. 

6 Metzen i Schaffel =6.1176 bushels. 

4 SchafTel i Muth =3.0588 quarters. 

Liquid Measure, 

I Maaskanne= .23529 gallon. 

64 Maaskannen i Eimer = 15.05856 gallons. 

25 Eimer i Fass =376.464 gallons. 

The Schenk-Eimer, ordinarily used in die Wine trade, contains only 
60 Maaskannen, equal to 14. 1 174 imperial gallons. 

V. Bavarian Measures of Weight. 

I Quentchen= .15433 ounce. 

4 Quentchen i Loth = .6173 ounce. 

32 Loth I Pfund = 1.23457 pounds. 

100 Pfund I Centner = [ ^^3-457 pounds, or 

( r.io2 hundredweights. 



WiJRTEMBEKG.— LENGTH, SURFACE, ETC. 165 

KINGDOM OF WURTEMBERG.— Old Weights and Measures.— 

Tables No. 47. 

I. WiJRTEMBERG MEASURES OF LENGTH. 

I Punkte = .01128 inch. 

10 Punkte I Linie = .1128 inch. 

10 Linien i ZoU = 1. 128 inches. 

10 Zoll I Fuss = .93995 foot 

10 Fuss I Ruthe = 9.3995 feet. 

2.144 Fuss I Elle = 2.015 feet 

6 Fuss I Klafter = 5.6397 feet 

26,000 Fuss I MeUe = i 8146.25 yards, or 

' ( 4.6285 miles. 

II. WiJRTEMBERG MEASURES OF SURFACE. 

1 Square Zoll = 1.272 square inches. 

100 Square ZolL i Square Fuss = -8835 square foot 

100 Square Fuss i Square Ruthe = 88.3506 square feet 

384 Square Ruthen... i Moigen = { 3769-626 square yards, or 

III. WiJRTEMBERG MEASURES OF VOLUME. 

I Cubic Linie = .001434 cubic inch. 

1000 Cubic Ltnien i Cubic Zoll = 1.434 cubic inches. 

1 000 Cubic Zoll I Cubic Fuss = . 83045 cubic foot 

144 Cubic Fuss I Cubic Klafter =119.583 cubic feet 

IV. WiJRTEMBERG MEASURES OF CAPACITY. 

I 

Dry Measure, 

I Viertlein = .305 pint 

4 Viertlein i Ecklein =1.219 pints. 

8 Ecklein i Vierling - 1.2 19 gallons. 

4 Vierling i Simri =4.876 gallons. 

8 Simri i Scheffel = 4.876 bushels. 

Liquid Measure, 

I Quart or Schoppen = .4043 quart. 

4 Quarts i Helleich Maass = 1.6173 quarts. 

I o Helleich Maass i Irai = 4-0433 gallons. 

16 Imi I Eimer = 64.6928 gallons. 

6 Eimer i Fuder =388.1568 gallons. 

V. WiJRTEMBERG MEASURES OF WEIGHT. 

I Quentchen = .1289 ounce. 

4 Quentchen i Loth . = .5156 ounce. 

32 Loth I Light Pfund = 1.03115 pounds. 

\V, S?.7huS*.."} ^ C»^- = -^^396 pounds. 
100 Light Pfund = 103. 115 pounds. 



1 66 WEIGHTS AND MEASURES. 

KINGDOM OF SAXONY.— Old Weights and Measures.— 

Tables No. 48. 

I. Saxon Measures op Length. 

I Linie = .07742 inch. • 

2 Linien i ZoU = .9291 inch. 

12 Zoll I Fuss = .9291 foot 

2 Fuss I EUe = 1.8582 feet 

2 Ellen I Stab = 3.7165 feet 

15 Fuss, 2 Zoll I Ruthe (Land Measure^ = 4.6972 yards. 

16 Fuss I Ruthe (Road Measure) = 4.9553 yards. 

I Lachter (Mining) = 2.1873 yards. 

1324.987 Ellen I Meile Post = 4.6604 miles. 

II. Saxon Measures of Surface. 

I Square Zoll = .8632 square. inch. 

144 Square Zoll i Square Fuss = .8632 square foot 

300 Square Ruthen i Acker = i . 4865 acres. 

III. Saxon Measures of Volume. 

I Cubic Zoll = .8021 cubic inch. 

1728 Cubic Zoll I Cubic Fuss = .8021 cubic foot 

108 Cubic Fuss I Klafter = 86.624 cubic feet 

3 Klafter i Schragen =259.873 cubic feet 

The. Klafter is 6 Fuss by 6 Fuss by 3 Fuss. The Schragen is used in 
the measurement of firewood. 

IV. Saxon Measures of Capacity. 

Dry Measure, 

I Maasche = 1.4463 quarts. 

4 Maaschen i Metze = 1.4463 gallons. 

4 Metzen i Viertel = 5.7852 gallons. 

4 Viertel i SchefTel = 2.8926 bushels. 

12 SchefFel i Malter =34.7124 bushels. 

2 Malter i Wispel = 69.4249 bushels. 

Liquid Measure, 

I Quartier = .2059 pint 

4 Quartier i Nossel = .8237 pint 

2 Nossel I Kanne = 1.6474 pints. 

36 Kannen i Anker = 7.4237 gallons. 

2 Anker i Eimer = 14.8262 gallons. 

3 Eimer i Oxhoft = 44.4687 gallons. 

6 Eimer i Fass or Barrel = 88.9374 gallons. 

V. Saxon Measures of Weight. 
The old Saxon measures of weight are the same as those of Prussia. 



BADEN. — LENGTH, SURFACE,- ETC 1 67 

GRAND DUCHY OF BADEN.— Old Weights and Measures.— 

Tables No. 49. 

I. Baden Measures of Length. 

I Punkte = .0118 inch. 

10 Punkte I Linie = .iiSi^jich. 

10 Linien i ZoU = 1. 181 inches. 

10 ZoU I Fuss = .9842 foot 

2 Fuss I EUe = 1.9685 feet 

10 Fuss I Ruthe = 9.8427 feet 

6 Fuss I Klafter = 5.9055 feet 

1 48 1 4.8 1 5 Fuss I Stunde =4860.59 yards. 

2 Stunden i Meile = 5.5234 miles. 

II. Baden Measures of Surface. 

I Square ZoU = i-395i square inches. 

100 Square ZoU i Square Fuss = .9688 square foot 

100 Square Fuss i Square Ruthe = 10.7643 square yards. 

100 Square Ruthen... i Viertel = 1076.43 square yards. 

4 ^i-tel X Morgan = { ^^^l^^S:/"'^' " 

III. Baden Measures of Volume. 

I Cubic Fuss = .95335 cubic foot 
X 44 Cubic Fuss i Klafter = 137.28 cubic feet 

IV. Baden Measures of Capacity. 

Liquid Measure. 

I Glass - 1.0563 gills. 

10 Glass I Maass = 1.3204 quarts. 

10 Maass .*. i Stutze = 3.3014 gallons. 

10 Stutzen I Ohm = 33.014 gallons. 

10 Ohm I Fuder = 330.14 gallons. 

Dry Measure. 

I Becher = .2643 pint 

10 Becher i Maasslein = .1652 peck. 

10 Maasslein i Sester = .4127 bushel. 

10 Sester i Maker = 4.1268 bushels. 

10 Malter i Zuber = 41.2679 bushels. 

V. Baden Measures of Weight. 

I As = .7716 grain. 

10 As I Pfennig = 7.716 grains. 

10 Pfennig...., i Centas = .1764 ounce. 

10 Centas i Zehnling = 1.7637 ounces. 

10 Zehnling i Pfund = 1.1023 pounds. 

100 Pfund iCentner = 110.230 pounds. 



1 68 WEIGHTS AND MEASURES. 

THE HANSE TOWNS. — OlD WEIGHTS AND MEASURES. — 

Tables No. 50. 

HAMBURG. — ^Weights and Measures. 

I. Hamburg Measures of Length. 

I Acht'el = .1175 inch. 

8 Achtel I ZoU = .9402 inch. 

12 ZoU I Fuss = . .9402 foot. 

2 Fuss i Elle = 1.8804 feet 

6 Fuss i Klafter, or Faden= 5.6413 feet 

14 Fuss i Marsch-Ruthe = 13.1629 feet 

16 Fuss I Geest-Ruthe = 15.0434 feet 

The Hamburg Elle above is used for silk, linen, and cotton goods. The 
Brabant Elle is equal to i V5 Hamburg Elle; and 4 of them are reckoned 
equal to 3 yards. The Prussian Ruthe is also used. The Prussian Fuss is 
used in surveying. 

II. Hamburg Measures of Surface. 

I Square ZoU = .8840 square inch, 

144 Square ZolL... i Square Fuss = .8840 square foot- 

196 Square Fuss... i Square Marsch-Ruthe = 173.26 square feet 

256 Square Fuss... i Square Geest-Ruthe = 226.30 square feet 

200 Square Geest- ) o^^effel Oe^t T^nd - i 5028.98 square yards, or 
Ruthen / ^ ^^^^^^ oeest-l^na - ^ ^^^^ ^^^^^ 

600 Sq. Marsch- ) ^ ^ ^ f ii55o-93 square yards, or 

Ruthen... J ^ ( 2.386 acres. 

III. Hamburg Measures of Volume. 

I Cubic Zoll = .8311 cubic inch. 

1728 Cubic 2k)ll I Cubic Fuss = .8311 cubic foot 

88.9 Cubic Fuss.... I (Cubic) Klafter = 73.88 cubic feet 
1 2 o Cubic Fuss i Tehr =99-73 cubic feet 

IV. Hamburg Measures of Capacity. 

Liquid Measure. 

I Ossel = .09965 gallon. 

2 Ossel I Quartier = .1993 gallon. 

2 Quartier i Kanne = .3987 gallon. 

2 Kannen i Stubchen =-• .7974 gallon. 

I Stubchen i Viertel = 1.5947 gallons. 

4 Viertel i Eimer ^ 6. 3 7 88 gallons. 

5 Viertel i Anker = 7.9735 gallons. 

6 Eimer i Tonne = 38.2 728 gallons. 

4 Anker i Ohm = 31.8940 gallons. 

6 Anker i Oxhoft = 47.8410 gallons. 

6 Ohm I Fuder, or Tonneau= 191.3640 gallons. 

The above are measures for Wines and Spirits. For Beer^ there are 
three sizes of Tonne, containing respectively 48, 40, and 32 Stubchen. 



HAMBURG. — WEIGHTS. 169 

Dry Measure. 

I Small Maass = .0236 bushel 

2 Small Maass i Large Maass = .0473 bushel. 

4 Laige Maass i Spint = .1890 bushel. 

4 Spint I Himten = .7560 bushel. 

2 Himten i Fass = 1.5121 bushels. 

2 Fass I Scheffel = 3.0242 bushels. 

10 Scheffeln i Wispel • = 30.2416 bushels. 

3 Wispel I Last =90.7248 bushels. 

For barley and oats, the Scheffel contains 3 Fass. 

V. Hamburg Measures of Weight. 

I Half Gramme = .0011 pound =.5 gramme. 

10 Half Grammen i Quint = .01102 pound =5 grammes. 

10 Quinten i (New) Unze = .11023 pound =50 „ 

10 (New) Unzen.. i (New) Pfund = i.io232pounds = 5oo „ 

100 (New) Pfund i Centner = 110.232 pounds = 50 kilog. 

60 Centners. i (Commercial) Last = { ^or^^ q^.? ^^^ =3000 kilog. 

This, it is apparent, is a metric system of weights, which was comparatively 
recently introduced and adopted at Hamburg. It is now, of course, over- 
ruled by the French metric system enforced for the German Empire. 



BREMEN. — Old Weights and Measures. 

The Fuss is equal to 11.392 inches, and the Klafter is equal to 5.696 
feet. The Morgen = .6368 acre. The principal measures for wines and 
spirits are the Viertel =1.56 gallons; the Anker = 5 Viertels = 7.80 gallons; 
the Oxhoft = 46.80 gallons. The Scheffel, for dry goods = 2.0388 bushels. 
The old weights are the same as those of Hamburg. 



LUBEC. — Old Weights and Measures. 

The Fuss is equal to 11.324 inches. The Viertel =1.60 gallons; the 
Anker = 8 gallons ; the Oxhoft = 48.04 gallons. The Scheffel, for dry goods, 
= .9545 bushel. The old Pfund =1.0725 pounds, and the Centner = 
1.0725 cwts. 



GERMAN CUSTOMS UNION.— Old Weights and Measures.— 

Table No. 51. 

Centner 110.23 pounds (50 kilogrammes). 

Ship-Last of timber about 80 cubic feet. 

Scheffel 1.512 bushels. 

Klafter 6 feet. 

In Oldenbuig, Hanover, Brunswick, Saxe-Altenbourg, Birkenfeld, Anhalt, 
Waldeck, Reuss, and Schaumburg-I^ippe, the old system of weights is the 
same as that of Prussia. 



I/O WEIGHTS AND MEASURES. 

AUSTRIAN EMPIRE. — WEIGHTS AND MEASURES. — ^Tables No. 52. 

I. Austrian Measures of Length. 

I Punkte = .0072 inch. 

12 Punkte I Linie = .0864 inch. 

12 Linien i 2k)ll = 1.0371 inches. 

12 Zoll I Fuss = 1.037 1 feet 

2 Fuss i EUe = 2.0742 feet 

6 Fuss I Klafter = 6.2226 feet 

4000 Klafter i Meile(post) = { ^^^J. J^^^. 

II. Austrian Measures of Surface. 

I Square Zoll = 1,0756 squareinches. 

144 Square Zoll i Square Fuss = 1.0756 square feet. 

36 Square Fuss i Square Klafter = | 38.7"5 square feet, or 

^ ^ -^uiu^ A ««Li.j ^1 4.3025 square yards. 

8 }4 Square Klafter, or ) « -d ^i. - o- -j 

300 Square ^uss } i Square Ruthe = 35.854 square yards. 

64 Square Ruthen i Metze = 2294.7 square yards. 

3 Metzen, or ) * , _ f 6884 square yards, or 

1600 Square Klafter J -^ " ( 1.4223 acres. 

III. Austrian Measures of Volume. 

Cudic Miosure, 

I Cubic Zoll = I. "55 cubic inches, 
1728 Cubic Zoll I Cubic Fuss = i-ii55 cubic feet 

216 Cubic Fuss.... i Cubic Klafter = [ "4^-94 cubic feet or 

\ 8.924 cubic yards. 

IV. Austrian Measures of Capacity. 

Dry Measure. 

iProbmetzen = ( '""SpinljOr 

( 3.665 cubic inches. 

8 Probmetzen i Becher = .8460 pint 

4 Becher i Futtermassel = 1.6920 quarts. 

2 Futtermassel i Muhlmassel = [ ^'tf ^T ^^' ^^ 

\ .8460 gallon. 

2 Muhlmassel i Achtel = 1.6920 gallons. 

2 Achtel I Viertel = [ 3-384o gallons, or 

( .4230 bushel. 

4 Viertel i Metze = 1.69 18 bushels. 

30 Meuen i Muth =/ So. 7536 bushels, or 

^ \ 8.3442 quarters. 



AUSTRIAN EMPIRE. — CAPACITY, WEIGHTS. 



171 



Liquid Measure. 
I Pfiflf 






2Pfiflf iSeidel = 

2 Seidel i Kanne = 

2 Kannen i Mass = 

10 Mass I Viertel = 

4 Viertel i Eimer = 

32 Eimer.... i Fuder = 



1.246 gills, or 
10.781 cubic inches. 
2.491 cubic inches, or 
.6229 pint 
1.2457 pints. 
1.2457 quarts. 
3. 1 143 gallons. 
12.4572 gallons. 
398.6304 gallons. 



V. Austrian Measures of Weight. 



{270.1 grains, or 
.6173 dram. 
2.4694 drams. 
9.8776 drams, or 
.6173 ounce. 
1.2347 ounces. 
4.9388 ounces. 
9.8776 ounces, or 
.6173 pound avoirdupois. 

= 1.2347 pounds avoirdupois. 

_ (123.47 pounds avoirdupois, or 
~ ( 1. 1024 hundredweights. 

In 1853, a pfimd of 500 grammes, with decimal subdivisions, was adopted 
tor customs and fiscal purposes. 



I Pfenning 

4 Pfenning i Quentchen 

4 Quentchen... i Loth 

2 Loth I Unze 

4 Unzen i Vierdinge 

2 Vierdinges... i Mark 
I.. I Pfund 
100 Pfimd I Centner 



2 Marks, or 
16 Unzen 






RUSSIA. — ^Weights and Measures. — ^Tables No. 53. 
L Russian Measures of Length. 

English Equivalent. 

I Vershok = 1.75 inches. 

16 Veishoks i Arschine = 28 „ 

3 Arschines i Sajene = 7 feet. 

( 3500 feet, or 

500 Sajenes i Verst = < ii66^ yards, or 

( 0.6629 mile. 

The Fuss, or Russian foot, is 13.75 inches; but, since 1831, the English 
foot of 1 2 inches has been used as the ordinary standard of length, each 
inch being divided into 12 parts. 



I Lithuanian Meile 5-5574 English miles. 

I Rhein Fuss, used in calculating ) t? r 1. r * 



export duties on timber 



/ 



1^2 WEIGHTS AND MEASURES. 

II. Russian Measures of Surface. 

o A I, .. f 784 square inches, or 

I Square Arschine = | ^ J ^ ^^^ ^^ 

9 Square Arschines.. i Square Sajene = | ^^.^J^u^S'y^s. 
2400 Square Sajenes i Desatine -\ ^' ; ^ o^^r^c ' 

III. Russian Measures of Capacity. 
Liquid Measure, 

I o Tscharkeys i Kruschka = 1.0820 quarts. 

1 00 Tscharkeys i Vedro = 2. 7049 gallons. 

3 Vedros i Anker = 8.1 147 „ 

tz v! AnkeS } •••• ' Sarokowaja Boshka = 108.196 

Dry Measure (Grain). 

I Gamietz = 2.885 quarts. 

2 Gamietz i Tschetwerka = i .4424 gallons. 

4 Tschetwerkas... i Tschetwerik = .7213 bushel. 

2 Tschetweriks.... i Pajak = 1.4426 bushels. 

2 Pajaks I Osmin = 2.8852 „ 

2 Osmins i Tschetwert* = 5.7704 „ 

■« T-he*e«s . L». = { ".ijfi^JSKL 

* A Tschetv^ert is usually reckoned as 5ji^ bushels, and 100 Tschetwerts as 72 quarters^ 
though they are more exactly 72.1308 quarters. 
100 quarters are equal to 138.637 Tschetwerts. 

For earthworks, masonry, &c., the Sajene is divided into tenths (dessiatka), 
hundredths (sotka), and thousandths (tisiatchka), which are used as a basis 
for lineal, superficial, and cubic measurements, similarly to the French 
metre with its sub-multiples. 

IV. Russian Measures of Weight. 



96 Dolls I Zolotnick = < 



I Dolis = .68576 grain. 

65.833 grains, or 
.1505 ounce. 
3 2k)lotnicks... i Lotti = .4514 » 

8 2k)lotnicks... i Lana = 1.2037 ounces. 

12 Lanas, or \ i .90285 pound avoirdupois, or 

32 Lottis, or > I Funt, or pound = < 14.446 ounces, or 
96 2k)lotnicks j (6320 grains. 

40 pounds I Pood = 36.114 pounds avoirdupois. 

10 Poods I Berkovite = \ 36^-^4 pounds avoiidupois. or 

( 3.224 hundredweights. 

3 Berkovitz i Packen = 9.672 hundredweights. 



HOLLAND, BELGIUM, NORWAY, ETC. I73 

62.0257 Poods I English ton. 

2481.0268 Russian pounds i „ 

The Pood is commonly estimated at 36 pounds avoirdupois. 
The Nurembeig pound, used for apothecaries' weight, weighs 5527 grams, 
or about .96 pound troy. 

The Ship-Last is equal to 2 tons. 

The Caraty for weighing pearls and precious stones, is about 3 ^6 grains. 



HOLLAND. 

The metric system was adopted in Holland in 1819; the denominations 
corresponding to the French are as follows : — 

Ltngth, — Millimetre, Streep; centimetre, Duim; decimetre, Palm; metre, 
El; decametre, Roede; kilometre, Mijle. 

Surface, — Square millimetre, Vierkante Streep; square centimetre, Vier- 
kante Duim; and so on. Hectare, Vierkante Bunder. 

Cubic MecLsure, — Millistere, Kubicke Streep, and so on. 

Capacity, — Centihtre, Vingerhoed; decilitre, Maatje; liquid litre, Kan; 
dry litre, Kop; decalitre, Schepel; Hquid hectolitre. Vat or Ton; dry 
hectolitre. Mud or Zak; 30 hectolitres = i Last= 10.323 quarters. 

Weight, — Decigramme, Korrel; gramme, Wigteje; decagramme, Lood; 
hectogramme, Onze; kilogramme, Pond. 



BELGIUM. 



The Frenc3i metric S)rstem is used in Belgium. The name Livre is 
sabstitated for kilogramme, Litron for litre, and Aune for metre. 



NORWAY AND DENMARK. 

Weights and Measures. — Tables No. 54. 

I. Norwegian and Danish Measures of Length. 

I Linie = .0858 inch. 

12 Linier i Tomme = 1.0297 inches. 

12 Tommer i Fod = 1.0297 feet 

2 Fod I Alen = 2.0594 „ 

3 Alen, or 
6 Fod 
2 Favn, or 

12 Fod 
2,000 Roder, or ) j^^ ^ i 8237-77 yards, or 



> I Favn = 6.1783 „ 

I Rode = 12.3567 „ 



24,000 Fod 



( 4.68055 miles. 



23,642 Fod I nautical mile= 4.61072 English miles. 

IL Norwegian and Danish Measures of Surface. 

144 Square Linie i Square Tomme = 1.0603 square inches. 

144 Square Tomme... i Square Fod = 1.0603 square feet 
144 Square Fod i Square Rode = 16.966 square yards. 



174 WEIGHTS AND MEASURES. 

III. Norwegian and Danish Measures of Volume. 

1728 Cubic Linier i Cubic Tomme = 1.0918 cubic inches. 

1728 Cubic Tomme.... i Cubic Fod = 1.0918 cubic feet 

The Favn of firewood measures 6x6x2 Fod= 72 cubic Fod = 78.60 
cubic feet In forest measure it is 6)4 x 6j4 x 2 Fod = 84)^ cubic Fod = 
92.26 cubic feet 

IV. Norwegian and Danish Measures of Capacity. 

Liquid Measure, 

I Paegle = .4248 pint 

4 Paegle i Pot = 1.6991 pints. 

2 Potter I Kande = 3.3983 „ 

38 Potter I Anker = 8.0709 gallons. 

136 Potter I Tonde = 28.885 * >> 

6 Ankeme i Oxehoved = 48.4256 „ 

4 Oxehoveder i Fad =193.7027 „ 

Dry Measure. 

I Pot = 1.699 1 pints. 

18 Potter I Skeppe = 3.8232 gallons. 

2 Skepper i Fjerdingkar = .9558 bushel. 

4 Fjerdingkar i Tonde = 3.8231 bushels. 

12 Tender i Laest =AS'^T^9 » 

V. Norwegian and Danish Measures of Weight. 

I Ort = 7.7163 grains. 

10 Ort I Kvint = 77.163 „ 

100 Kvinten i Pund = 1.1023 pounds. 

100 Pund I Centner =110.23 „ 

40 Centner i Last = i . 9684 tons. 

52 Centner i Skip-Last = 2.5590 „ 

16 Pund I Lispund = 17.63 7' pounds. 

320 Pund I Skippund = 3.149 cwts. 



SWEDEN. — Weights and Measures. — Tables No. 55. 

I. Swedish Measures of Length. 

I Linie = .1169 inch. 

10 Linier i Turn = 1. 1689 inches. 

10 Tumer i Fot = 11.6892 „ 

10 Fot I Stang = .9.7411 feet 

10 Stanger i Ref = 32.4703 yards. 

760 Ref I Meile - / ">6S9-3o8 yards, or 

^^"^ ^^* ' ^"^^ - \ 6.6417 miles. 

2 Fpt I Aln = 1-942 feet 

6 Fot I Faden =^ 5.845 „ 



SWEDEN, SWITZERLAND. 1 75 

II. Swedish Measures of Surface. 

loo Square Linier... i Square Turn = 1.3666 square inches. 

100 Square Turner., i Square Fot = -9489 square foot 

1 00 Square Fot i Square Stang = 3* 5 1 46 square yards. 

100 Square Stanger i Square Ref = I •°54-^ square yards, or 

4 Square Fot i Square Aln = 3-7956 square feet. 

5.6 Square Ref:... r Tunnlar^d = { "^Lp^^S"'" " 

III. Swedish Measures of Volume. 

Cttbic Measure, 

I Cubic Turn =1.5972 cubic inches. 

1000 Cubic Turner i Cubic Fot = .9263 cubic foot. 

8 Cubic Fot I Cubic Aln = 7.4104 cubic feet 

Liquid and Dry Measure. 

1 000 Cubic Linier i Cubic Turn = . 1 843 gill. 

1 00 Cubic Turner i Kanna = 2. 3096 quarts. 

10 Kanna i Cubic Fot = 5.774 gallons. 

8 Cubic Fot I Cubic Aln =46.192 „ 

IV. Swedish Measures of Weight. 

I Kom = .6564 grain. 

100 Kom I Ort = 2.4005 drams. 

100 Ort I Skalpund = .9377 pound. 

-o Skalpur^d X Cer^tner = { 93-7739 pounds, or 

IOC Centner i Ny-Last = 4.1892 tons. 

A Pund, commercial, is .9377 pound. 

A Pund, freight, is .75016 pound. 

A Pund, miners* mark weight, is .8285 pound. 

A Pund, country town's mark weight, is .7891 pound. 



S^VITZERLAND. — ^WEIGHTS AND MEASURES. — Tables No. 56. 

I. Swiss Measures of Length. 

I Striche = .01181 inch. 

10 Striche I Linie = .11811 „ 

10 Linien i 2k)ll = 1.181 12 inches. 

10 Zoll I Fuss = 11.81124 „ 

2 Fuss i Elle = 1.9685 feet 

6 Fuss i Klafter = 59056 ^ 

10 Fuss i Ruthe = 98427 „ 

1600 Ruthen .... i Schweizer-stunde, or Lien= < 2 o82^^iles 



176 WEIGHTS AND MEASURES. 

II. Swiss Measures of Surface. 

I Square ZoU = 1-3947 square inches. 

100 Square ZoU i Square Fuss = .9688 square foot 

36 Square Fuss i Square Klafter = 34.8768 square feet 

100 Square Fuss i Square Ruthe = 10.7643 square yards 

400 Square Ruthen. . i Juchart = .8694 acre. 

6400 Jucharten i Square Stunde = 5693.52 acres. 

350 Square Ruthen i Juchart, of meadow land. 

450 Square Ruthen i Juchart, of woodland. 

III. Swiss Measures of Volume. 

I Cubic Zoll = 1.6476 cubic inches. 

1000 Cubic ZolL I Cubic Fuss = .9535 cubic foot 

216 Cubic Fuss I Cubic Klafter = 7.6172 cubic yards. 

1000 Cubic Fuss I Cubic Ruthe =35.3166 „ 

IV. Swiss Measures of Capacity. 

Dry Measure, 

I Imi = 1.3206 quarts. 

10 Imi I Maass = .4127 bushel. 

10 Maass i Malter = 4.1268 bushels. 

Liquid Measure, 

2 Halbschoppen i Schoppen = 2.6412 gills. 

2 Schoppen i Halbmaass = 1.3206 pints. 

2 Halbmaass i Maass = 2.6412 „ 

100 Maass i Saum = 33.015 gallons. 

V. Swiss Measures of Weight. 

I Quntii = 2.2048 drams. 

4 Quntli I Loth = .5511 ounce. 

2 Loth I Unze = 1. 1023 ounces. 

16 Unzen i Pfund = 1. 1023 pounds. 

100 Pfund I Centner = 110.233 pounds, or .9842 cwt. 

The Pfund is divided into halves, quarters, and eighths. It is also 
divided into ^00 Grammes, and decimally into Decigrammes, Centi* 
grammes, and Milligrammes. 



SPAIN. — ^Weights and Measures. — Tables No. 57. 

The French metric system was established in Spain in 1859. The metre 
is named the Metro; the litre, Litro; the gramme, Grammo; the are, Area; 
the tonne, Tonelada. The metric system is established likewise in the 
Spanish colonies. The old weights and measures are still largely used. 



SPAIN — LENGTH, SURFACE, ETC. 1 77 

L Old Spanish Measures of Length. 

I Punto = .00644 inch. 

12 Puntos I Linea = .07725 inch. 

12 Lineas i Pulgada = .927 inch. 

6 Pulgadas i Sesma = 5.564 inches. 

2 Sesmas i Pies de Burgos = .92 73 foot. 

3 Pies de Burgos i Vara = 2.782 feet 
2 Varas i Estado = 5.564 feet 

4 Varas i Estadal =11.128 feet 

5000 Varas i Legua (Castilian) = 2.6345 miles. 

8000 Varas i Legua (Spanish) = 4.2 151 miles. 

II. Old Spanish Measures of Surface. 

I Square Pies = .860 square foot 

9 Square Pies i Square Vara = .860 square yard. 

16 Square Varas i Square Estadal = 13.759 square yards. 

50 Square Varas .... i Estajo =42.997 square yards. 

576 Square Estadals. i Fanegada = 1.6374 acres. 

50 Fanegadas. i Yugada = 81.870 acres. 

III. Old Spanish Measures of Capacity. 

Liquid Measure. 

I Capo = .888 gill. 

4 Capos I Cuartillo = .111 gallon. 

4 Cuartillos i Azumbre = .444 gallon. 

2 Azumbres i Cuartilla = .888 gallon. 

^ ^.« ( I ArrobaMayor,orCantara) ^ ^^ „«n^«« 
4 CuartUlas... | (for wine) j " 3-552 gallons. 

16 Cantaras i Mayo =56.832 gallons. 

The old measure for oil is the Arroba Menor= 2.7652 gallons. 

JDry Measure, 

I Ochavillo = .00785 peck. 

4 Ochavillos i Racion = .0314 peck. 

4 Raciones i Quartillo = . 03 1 4 bushel. 

2 Quartillos i Medio = .0628 bushel. 

2 Medios i Almude = .1256 bushel; 

12 Amuerzas i Fanega = 1.5077 bushels. 

1 2 Fanegas i Cahiz = 1 8.0920 bushels. 

IV. Old Spanish Weights. 

I Grano = .771 grain. 

12 Granos i Tomin = 9.247 grains. 

3 Tomines .... I Adarme = 27.74 grains. 

2 Adarmes .... i Ochavo, or Drachma = .1268 ounce. 

8 Ochavos i Onza = i.o 144 ounces. 

8 Onzas i Marco = 8.1 154 ounces. 

2 Marcos i Libra (Castilia a) = i . o 1 44 pounds. 

IOC Libras i Quintal = 101.442 pounds. 

10 Quintals i Tonelada =1014.42 pounds. 

19 



178 WEIGHTS AND MEASURES. 



PORTUGAL. 



The French metric system of weights and measures v^as adopted in its 
entirety during the years 1860-63, and was made compulsory from the ist 
October, 1868, The chief old measures still in use are, the Libra = 1.012 
pounds; Almude, of Lisbon = 3.7 gallons; Almude, of Oporto = 5.6 gallons ; 
Alquiere = 3.6 bushels; Moio = 2.78 quarters. 



ITALY. 



The French metric system is used in Italy. The metre is named the 
Metra; the are, Ara; the stfere, Stero; the litre, Litro; the gramme, Gramma; 
the tonneau m^trique, Tonnelata de Mare. The various old weights and 
measures of the different Italian States are still occasionally used. 



TURKEY. 

Length. — i Pike or Dri= 27 inches, divided into 24 Kerats; i Forsang 
= 3.116 miles, divided into 3 Berri; the Surveyor's Pik, or the Halebi 
= 27.9 inches; and 5J^ Halebis= i reed. 

Surface, — The squares of the Kerat, the Pike, and the Reed. The 
Feddan is an area equal to as much as a yoke of oxen can plough in a 
day. 

Capacity^ Dry, — ^The Rottol-i.411 quarts, contains 900 Dirhems; 
22 Rottols= I Killow= 7.762 gallons, or .97 bushel, the chief measure for 
grain. 

Liquid. — i Oka= 1.152 pints; 8 Oke= i Almud= 1.152 gallons; i Rottol 
= 2.5134 pints; 100 Rottols= i Cantar = 31.417 gallons. 

Weights. — ^The Oke= 2.8342 pounds, divided into 4 Okiejehs, or 400 
Dirhems of 1.81 drams; i Rottolo= 1.247 pounds; 100 Rottolos= i Cantar 
= 124.704 pounds. 



GREECE AND IONIAN ISLANDS. 

The French metric system is employed in Greece. The metre is named 
the Pecheus; kilometre, Stadion; are, Stremma; litre, Litra; gramme. 
Drachm^. 1% kilogrammes = i Mni; lyi Quintals = i Tolanton 
i}i Tonneaux= I Tonos= 29.526 cwts. 

In the Ionian Islands, whilst they were under the protection of Great 
Britain (1830 to 1864), the British weights and measures were those in use, 
with Italian names. The foot was named the Piede; the yard, the Jarda; 
the pole, the Camaco; the furlong, the Stadio; the mile, the Miglio. The 
gallon was the Gallone; the bushel, the Chilo; the pint, the Dicotile; the 
pound avoirdupois, the Libra Grossa; the pound troy, the Libra Sottile. 
The Talanto consisted of 100 pounds, and the Miglio of 1000 pounds. 



MALTA. 



In round numbers, 3^ Palmi= i yard; i Canna = 2 "/^ yards. 
The Salma = 4.964 acres. Approximately, 543 Square Palmi = 400 
square feet; 16 Salmi = 71 acres. 



EGYPT. — LENGTH, SURFACE, ETC. 1 79 

I Cubic Tratto = 8 cubic feet; 144 Cubic Palmi = 96 cubic feet; i Cubic 
Canna = 543 cubic feet 

Approximate weights: — 15 Oncie=i4 ounces; i Rotolo=i^ pounds; 
4 Rotoli = 7 pounds; 64 Rotoli = i cwt. ; i Cantaro =175 pounds; i Quintal 
= 199 pounds; 64 Cantari = 5 tons. 



EGYPT. — ^Weights and Measures. — ^Tables No. 58. 
I. Egyptian Measures of Length. 

Pik, or cubit of the Nilometre 20.65 inches. 

Pik, indigenous 22.37 „ 

Pik, of merchandise 25.51 „ 

Pik, of construction 29.53 „ 

6 Palms I Pik. 

24 Kirats i Pik or Dr^, 

4.73 Piks of construction... i Kassaba in surveying, =11.65 feet. 

11. Egyptian Measures of Surface. 

I Square Pik = 6.055 square feet 

22.41 Square Piks i Square Kassaba = 15.07 square yards 

333.33 Square Kassaba, i Feddan = .9342 acre. 

III. Egyptian Measures of Capacity. * 

I Kadah = 1.684 pints. 

2 Kadahs i Milwah = 6.735 » 

2 MDwahs I Roobah = 1.684 gallons. 

2 Roobahs '. i Kelah = 3.367 „ 

2 Kelehs i Webek = 6.734 „ 

6 Webeks i Ardeb - / ^0.404 gallons, or 

^ ^^«^^*^ ^ ^^^^ - I 6.48 cubic feet 

The Guirbah of water (a government measure) is V15 cubic metre = 66*4 
litres, or 11.772 cubic feet 

IV. Egyptian Measures of Weight. 

I Kamhah - .746 grain. 

4 Kamhahs. i Kerat 

16 Kerats i Dirhem = 1.792 drachms. 

24 Kerats i Mitkal. 

SMitkals i Okieh. 

x^: Salt's"} ^R°"°l = -^S.! pound. 

100 Rottols I Kantar =98.207 pounds. 



400 Dirhems i Oke = 2.728 „ 

36 Okes I Kantar =98.207 „ 



l80 WEIGHTS AND MEASURES. 



MOROCCO. 



Length, — ^The Tomin = 2.81025 inches; the Dra'a = 8 Tomins = 22.482 
inches. 

Capacity, — ^The Muhd = 3.08135 gallons; the Sai = 4 Muhds= 12.3254 
gallons. 

Weights, — The Uckia = 392 grains; the Rotal or Artal = 2o Uckieh = 
1,12 pounds; the Kintar= 100 Rotales= 112 pounds. 

Oil is sold by the Kula = 3.3356 gallons. Other liquids are sold by 
weight 



TUNIS. 

Length, — ^The Dhrai, or Pike, is the unit of length. The Arabian Dhraa, 
for cotton goods =19.224 inches; the Turkish Dhrai, for lace = 25.0776 
inches; the Dhrai Endaseh, for woollen goods = 26.4888 inches. 

The Mil Sah'ari = .9i49 mile. 

Capacity, — For dry goods the Sai= 1.2743 pint; 12 Saa=i Hueba = 
6.8228 gallons. 

For liquids, the Pichoune = .4654 pint; 4 Pichounes=i Pot =1.8616 
pints; 15 Pots = I Escandeau, and 4 Escandeaux=i Mill^role= 13.9623 
gallons. 



ARABIA. 
The weights" and measures of Egypt are used in Arabia. 



CAPE OF GOOD HOPE. 

The standard weights and measures are British, with the excepfton of the 
land measure. To some extent, the old British and the Dutch measures 
are in use. The general measure of surface is the old Amsterdam Morgen^ 
reckoned equal to 2 acres; though the exact value is equal to 2.1 1654 
acres. 1000 Cape feet are equal to 1033 British feet 



INDIAN EMPIRE. — ^WEIGHTS AND MEASURES. 

An Act " to provide for the ultimate adoption of an uniform system of 
weights and measures of capacity throughout British India " was passed in 
October, 187 1. The ser is adopted under the Act as the primary standard 
or unit of weight, and is a weight of metal in the possession of the Govern- 
ment, equal, when weighed in a vacuum, to one kilogramme. The unit of 
capacity is the volume of one ser of water at its maximum density, equiva- 
lent to the litre. Other weights and measures are to be multiples or sub- 
multiples of the ser, and of the volume of one ser of water. 

The following are the weights and measures in common use in India: — 



BENGAL — LENGTH, SURFACE, ETC l8l 

BENGAL. — ^Weights and Measures. — Tables No. 59. 

I. Bengal Measures of Length. 

I Jow, or Jaub = ]^ inch. 

3 Jow I Ungulee = ^ „ 

4 Ungulees i Moot = 3 inches. 

3 Moots I Big'hath, or Span = 9 „ 

2 Big'haths i Hit'h, or Cubit... = 18 „ 

2 Hat'h I Guz = I yard. 

2 Guz I Danda, orFathom = 2 yards. 

^o- ^^^ ^ Coss = { --15t'i. 

4 Coss I Yojan = 4.5454 miles. 



II. Bengal Measures of Surface. 

I Square Hat'h = 2.25 square feet. 

4 Square Hit'hs i Cowrie = i square yard. 

4 Cowries i Gunda = 4 square yards. 

20 Gundas i Cottah = 80 „ 

20 Cottahs I Beegah = i '^°° ^""f^ V^^' °' 

\ -3306 acre. 

For land measure, the following table is used for Government surveys : — 

I Guz = 33 lineal inches. 

3 Guz I Baus,orRod= 8^ lineal feet. 

9 Square Guz i Square Rod = 68 Vx6 square feet. 

400 Square Rods..... i Beegah = { 3°*S Weyards, or 

III. Bengal Measures of Capacity. 

The Seer is a measure common to liquids and dry goods. It is taken 
at 68 cubic inches, or 1.962 pints, in volume. But it varies in different 
localities. 5 Seer= i Palli, and 8 Palli= i Maund, or 9.81 gallons. The 
Sooli = 3.065 bushels, and 16 Soolis = i Khahoon, or 49.05 bushels. 

IV. Bengal Measures of Weight. 

The Tola, or weight of a Rupee, 180 grains, is the unit of weight. 

I Tola =180 grains. 

5 Tolas I ChittHk =900 „ 

16 Chittiks I Seer = 2.057 pounds. 

5 Seers i Passeeree = 10.286 „ 

8 Passeerees i Maund = 82.286 „ 



MADRAS. — Weights and Measures. — Tables No. 60. 

I. Madras Measures of Length. 

The English foot and yard are used. The Guz is 33 inches. The Baum 
or fathom is about 6j4 feet. A Nilli-Valli is a little under ij^ miles. 
7 Naili-Valli = I Kadam, or about 10 miles. The following are native 
measures : — 



1 82 WEIGHTS AND MEASURES. 

8 Torah i Vurruh = .4166 inch. 

24 Vurmh i Mulakoli = 10 inphes. 

4 Mulakoli i Dumna = 40 „ 

II. Madras Measures of Surface. 

The English acre is generally known. The native me^tsures are uncer- 
tain. In Madras and some other districts^ the following native measures 
are used : — 

I Coolie = 64 square yards. 

4 Ve Coolies i Ground = 266^ square yards. 

24 Grounds, or ) rown*** - / 6400 square yards, or 

100 Coobes J (1.3223 acres. 
16 Annas (each 400 yards), i Cawnie. 

III. Madras Measures of Capacity. 

I OUuck = .361 pint 

8 OUucks I Puddee = 1.442 quarts. 

8 Puddees i Mercil = 2.885 gallons. 

5 Mercils i Parah =14.426 „ 

80 Parahs i Garce = 18.033 quarters. 

This, though the legal system, is not used. The "customary" Puddee is 
still in general use; it has, when slightly heaped, a capacity of 1.504 quarts. 
The Mercil has a capacity of 3.0006 gallons; but, when heaped, it is equal 
to 8 heaped Puddees. The Seer-measure is the most common; its cubic 
contents are from 66 }4 to 67 cubic inches. 

IV. Madras Measures of Weight. 

I Tola = 180 grains. 

3 Tolas I PoUum = 1.234 ounces. 

8 PoUums I Seer = 9-874 „ 

5 Seers i Viss = 3.086 pounds. 

8 Viss I Maund = 24.686 „ 



20 



Maunds . Candy = { ^^J J^^ J^ '^' " 



In commerce, the Viss is reckoned as 3^ pounds; the Maund, 25 
pounds; and the Candy, 500 pounds. 



BOMBAY. — ^\Veights and Measures. — Tables No. 61. 

I. Bombay Measures of Length. 

I Ungulee = 9/,6 inch. 

2 Ungulee i Tussoo = i^ inches. . 

8 Tussoos I Vent'h =9 „ 

16 Tussoos I Hat'h =18 „ 

24 Tussoos I Guz =27 „ 

The Builder's Tussoo = 2.3625 inches in Bombay; and i inch in Surat 



BOMBAY, CEYLON, BURMAH. 1 83 

II. Bombay Measures of Surface. 

34 Ve Square Hat'h... i Kutty = 9-3i75 square yards. 

20 Kutties I Fund = 196.35 „ 

-P-nd ^ Beegah= {39^7^f-yf «'- 

1 20 Beegah i Chahur = 97-368 acres. 

In the Revenue Field Survey, the English acre is used 

III. Bombay Measures of Capacity. 

I Tippree= .2800 pint. 

2 Tipprees i Seer = .5600 „ 

4 Seers i Pylee = 2.2401 pints. 

16 Pylees i Parah = 4.4802 gallons. 

8 Parahs i Candy = 35.8415 „ 

25 Pamhs I Mooda = { '""^^^S g^Uons or 

•^ ( I-750I quarters. 

Another liquid measure is the Seer of 60 Tolas = 1.234 pints. 
In timber measurement in the Bombay dockyards, a Covit or Candi = 
12.704 cubic feet. 



CEYLON. 
The British weights and measures are used. 



BURMAH. 



The English yard, foot, and inch are being adopted; also the English 
Measures of Capacity. Weights. — The Piakthah or Viss is 3.6 pounds, and 
contains 100 Kyats of 252 grains each. 



CHINA. — ^Weights and Measures. — Tables No. 62. 

I. Chinese Measures of Length. 

I Fun = .141 inch. 

10 Fun I Tsun = i. 41 inches. 

10 Tsun I Chih =14.1 „ 

10 Chih I Chdng= 11.75 ^"^^t. 

10 Ching I Yin =39.17 yards. 

The Chih of 1 4.1 inches is the legal measure at all the ports of trade. 
At Canton, the values of the Chih are as follows : — 

Tailor's Chih 14.685 inches. 

Mercer's Chih (wholesale) 1 4. 66 to 1 4. 7 2 4 inches. 

Mercer's Chih (retail) ^4*37 to 14.56 „ 

Architect's Chih 12.7 inches. 

At Pekin there are thirteen different Chihs. 






1 84 WEIGHTS AND MEASURES. 

Distance, 

5 Fun I Li = .486 inch. 

10 Li )^Chih = .405 foot. 

5 Chih I Pii = 4.05 feet. 

360 Pii I Li =486 yards. 

250 Lf I Tii (or Degree) = 69 miles. 

II. Chinese Measures of Surface. 

25 Square Chih i Pii or Kung= 3.32 square yards 

60 Kung I Kish =199.47 

4 Kish I Mau =797-89 

100 Mau I King = 16,485 acres. 

The chief land measure is the Mau, than which smaller areas are 
expressed decimally. 

III. Chinese Measures of Capacity. {Dry Measure,) 

I Koh = .0113 gallon. 

5 Koh >^Shing = .0565 „ 

10 Koh I Shing = .113 „ 

10 Shing I Tau =1.13 gallons. 

Liquids are measured by vessels containing definite weights, as i, 2, 4^ 
and 8 Taels; also large earthen vessels holding 15, 30, and 60 Catties. 
See Table of Weights. 

IV. Chinese Measures of Weight. 

I L^ang or Tael = i '/a ounces. 
16 L^ang I Kin or Catty = i V3 pounds. 

100 fcin I Tan or Pecul = [ ^33-33 Pounds, or 

( 1. 19 cwts. 



COCHIN-CHINA. 



Length, — The Thuoc, or cubit, 19.2 inches, is the chief unit of measure 
of length. It varies considerably for different places. The Li or mile is 
486 yards; 2 Li make i Dam; and 5 Dam make i league =2.761 miles. 

Surface, — 9 Square Ngu make i Square Sao = 64 square yards. 100 
Square Sao make i Square Mao = 6400 square yards, or 1.32 acres. 

Weights, — The smallest weight is the Ai = .0000006 grain. The weights 
ascend by a decimal scale, until 10,000,000,000 Ai are accumulated = 
I Nen = .8594 pound. The greatest weight is the Quan = 6875^ pounds. 

Capacity for Grain, — i Hao = 6»/g gallons. 2 Hao = i Shita=i2 4/j 
gallons. 



PERSIA. 



Length, — The Gereh = 2^ inches; 16 Gerehs= i Zer=38 inches. The 
Kadam or Step = about 2 feet; 12,000 Kadam = i Fersakh = about 4)^ miles. 



PERSIA, JAPAN. 185 

Surfcue and Cubic Measures, — ^These are the squares and cubes of the 
lengths. 

Capacity (Dry Goods). — The Sextario = .o7236 gallon. 4 Sextarios = 
I Chenica; 2 Chenicas= i Capicha; 3^ Capichas= i CoUothun; 8 Collo- 
thun = I Artata= 1.809 bushels. 

liquids are sold by weight 

Weights, — ^The Miscal = 7i grains; 16 Miscals=i Sihr; 100 Miscals = 
I Ratal = 1.014 pounds; 40 Sihrs = i Batman (Maund) = 6.49 pounds; 100 
Batman (of Tabreez)= i Karwar = 649.i42 pounds. 



JAPAN. — ^Weights and Measures. — Tables No. 63. 

I. Japanese Measures of Length. 

I Rin = .012 inch. 

10 Rin I Boo = .120 inch. 

10 Boo I Sun = 1.20 inches. 

10 Sun I Shiaku = i foot. 

3 Shiaku ^ Ken = i yard. 

6 Shiaku i Ken - 2 yards. 

60 Ken I Chu = 120 yards. 

_ J 4320 yards, or 
2.454 miles. 



36 Chu., I Ri = I 



Rough timber is sold by the Yama-Ken-Zau = d^ Sun. Cloth is measured 
by the Shiaku of 15 inches, with decimal sub-multiples. 

II. Japanese Measures of Surface. 

I Po = 4 square yards. 

30 Po I Is'she = 120 square yards. 

I o Is'she I It'tau = 1 200 square yards. 

10 Iftau. I Ifchoe = ( "°°° T"^ y^''*'' °'" 

I 2.4793 acres. 

The square Ken is the unit of square measure, equal to 4 square yards, 

III. Japanese Measures of Capacity. 

* Dzoku = .0000328 pint. 

10 Dzoku I Ke = .000328 pint. 

10 Ke I Sat = .000328 pint. 

10 Sats I Sai = .00328 pint. 

10 Sal I Shiaku = .03283 pint. 

10 Shiaku i Goo = .3283 pint 

10 Goo I Shoo = .4104 gallon. 

10 Shoo I To = 4.104 gallons. 

xo To I Koku =41.04 gallons. 



1 86 WEIGHTS AND MEASURES. 

IV. Japanese Measures of Weight. 

I Mo = .027 grain. 

10 Mo I Rin = .2701 grain. 

10 Rin I Fun = 2.701 grains. 

10 Fun I Noihme = 27.006 grains. 

4 Nomme i Riu = 108.026 grains. 

40 Riu I Kiu = .6173 pound. 

JAVA. 

Length. — The Duim=i.3 inches. 12 Duims=i foot. The Ell = 
27.08 inches. 

Surface, — The Djong of 4 Bahu = 7.015 acres. 

Capacity, for rice and grain. — The measures are in fact measures of 
definite weights, i sack = 61.034 pounds; 2 sacks =1 Pecul; 5 Peculs 
= I Timbang = 5.45 cwts.; 6 Timbang= i Coyaii = 32.7 cwts. For Hquids: 
The Kan = .328 gallon; 388 Kans=: 1 Leager= 127.34 gallons. 

Weights. — The Tael = 593.6 grains; i6Taels=^i Catty = 1.356 pounds: 
100 Catties = i Pecul- 135.63 pounds. 



UNITED STATES OF AMERICA. 

Length, — The measures are the same as those of Great Britain. 

In Land Surveying, the unit of measurement is the chain, and it is deci- 
mally subdivided. 

In City Measurements, the unit is the foot, and it is decimally subdivided. 

In Mechanical Measurements, the unit is the inch, and it is divided into 
a hundred parts. 

Surface, — The measures are the same as those of Great Britain. 

Capacity, — The measures of capacity for dry goods and for liquids are the 
same as the old English measures. 'Fhe standard U. S. gallon is equal to 
the old English wine gallon, or 231 cubic inches; it contains 8^ pounds 
of pure water at 62° F. 

Dry Measure, — Table No. 64. 

I gill. = .96945 imperial gill. 

4 gills I pint = .96945 imperial pint 

2 pints I quart ~ 1.9388 „ pints. 

4 quarts i gallon = .96945 „ gallon. 

2 gallons I peck = 1.9388 „ gallons. 

4 pecks I bushel = -96945 „ bushel. 

4 bushels I coomb =3-8777 „ bushels. 

2 coombs I quarter = .96945 „ quarter. 

5 quarters i wey or load =4.8472 „ quarters. 

2 weys I last =9-6945 „ quarters. 

For the Wine and Spirit Measures, and the Ale and Beer Measures, see 
the Old Measures of Great Britain, page 139. 

I cord of wood =128 cubic feet = (4 feet x 4 feet x 8 feet). 

Weights, — The Weights are the same as those of Great Britain. (See 
page 140.) 



BRITISH NORTH AMERICA, ETC. 1 8/ 

There are, in addition, the Quintal or Centner of loo pounds; and the 
New York ton of 2000 pounds, which is also used in most of the States, 
The old hundredweight and ton are for the most part superseded by the 
quintal and the New York ton. 

The French metric system of weights and measures has been legalized 
concuirendy with the old system. 



BRITISH NORTH AMERICA.— WEIGHTS AND MEASURES. 

Until the 23d May, 1873, the standard measures of length and surface, 
and the weights, were the same as those of Great Britain; whilst the 
measures of capacity were the old British measures for dry goods, for wine, 
and for ale and beer. At the above-named date a new and uniform system 
of weights and measures came into force, in which the imperial yard, pound 
avoirdupois, gallon, and bushel, became the standard units, and the 
imperial system was adopted in its integrity, with two important exceptions : 
that the hundredweight of 112 pounds, and the ton of 2240 pounds were 
abolished; and the hundredweight was declared to be 100 pounds, and the 
ton 2000 pounds avoirdupois, — thus assimilating the weights of Canada to 
those of the United States. 

The French metric system of weights and measures has been made 
permissive concurrently with the standard weights and measures. 



MEXICO. 
The weights and measures are the old weights and measures of Spain. 



CENTRAL AMERICA AND WEST INDIES. 

WEST INDIES (British). 
The weights and measures are the same as those of Great Britain. 

CUBA. 

The old weights and measures of Spain are in general use. For engineer- 
ing and carpentry work the Spanish, English, and French measures are in 
use. The French metric system of weights and measures is legalized, and 
is used in the customs departments. 

GUATEMALA AND HONDURAS. 
The weights and measures are the old weights and measures of Spain. 

BRITISH HONDURAS. 
In British Honduras, the British weights and measures are in use. 

COSTA RICA. 

The old weights and measures of Spain are in general use. But the 
introduction of the French metric system is contemplated. 



l88 WEIGHTS AND MEASURES. 

ST. DOMINGO. 

The old Spanish weights and measures are in general use. The French 
metric system is coming into use. 

SOUTH AMERICA. 

COLOMBIA. 

The French metric system was introduced into the Republic in 1857, 
and is the only system of weights and measures recognized by the govern- 
ment. In ordinary commerce, the Oncha, of 25 lbs., the Quintal, of 
100 lbs., and the Carga, of 250 lbs., are generally used. The libra is 
1. 102 pounds. The yard is the usual measure of length. 

VENEZUELA. 
The system and practice are the same as those of Colombia. 

ECUADOR. 

The French metric system became the legal standard of weights and 
measures on the ist January, 1858. 

GUIANA. 

In British Guiana, the weights and measures are those of Great Britain. 
In French Guiana or Cayenne, the ancient French system is practised In 
Dutch Guiana, the weights and measures of Holland are employed. 

BRAZIL. 

The French metric system, which became compulsory in 1872, was 
adopted in 1862, and has since been used in all official departments. But 
the ancient weight§ and measures are still partly employed. They are, with 
some variations, those of the old system of Portugal. 

Length, — ^The Line = .09 11 inch, and is divided into tenths. The PoUe- 
gada = 1.0936 inches. The Pd = 1 3. 1 236 inches, or ^j^ metre. The Vara = 
1.2 1 5 yards; and ij^ Varas = the geometrical pace =1.8227 yards. The 
Milha= 1.2965 miles; and 3 Milhas = i Legoa = 3.8896 miles. 

6 yards are reckoned equal to 5 Varas. 

Surface, 

64 Square Pollegadas... i Square Palmo = .5315 square foot 

25 Square Palmos i Square Vara =1.4766 square yards. 

4 Square Varas i Square Braga = 5. 9063 „ 

4840 Square Varas i Geira = 1.4766 acres. 

Capacity {Dry Goods), — The Salamine = .38o8 gallon; 2 Salamines = 
\ Oitavo; 2 Oitavo = i Quarto; 4 Quartas=i Alqueiro = .38o8 bushel; 
4 Alqueiras = i Fangas; 15 Fangas = i Moio = 2.8560 quarters. 

Liquids, — The Quartilho = .6i4i pint; 4 Quartilhos= i Canada; 6 Cana- 
das = I Pota or Cantaro; 2 Potas = i Almuda = 3.6846 gallons. 



PERU, CHILI, BOLIVIA, ETC. 1 89 

Weights. — The Arratel= 1.0119 pounds, is divided into 16 Ongas, and 
then into 8 Oitavos. 32 Arratels=i Arroba; 4 Arrobas = i Quintal = 
129.5181 pounds; and 13 J^ Quintals = 1 Tonelada= 15.6116 cwts. 

There is also the Quintal of 100 Arratels; Ships' freight is reckoned by 
the English ton = 70 Arrobas. 

PERU. 

The French metric system was established in i860, but is not yet gener- 
ally used. The weights and measures in common use are : — ^The ounce = 
1.014 ounce; the Libra=i.oi4 pound; the Quintal = 101.44 pounds; ^^ 
Arroba =25.36 pounds, or 6.70 gallons; the gallon = .74 imperial gallon; 
the Vara = .92 7 yard; the square Vara = .85 9 square yard. 

CHILL 

The French metric system has been legally established; but the old 
weights and measures are still in general use. These are the same as those 
of Peru. 

BOLIVIA. 

The weights and measures are the same as the old weights and measures 
of Peru and Chili. 

ARGENTINE CONFEDERATION. 

The French metric system has recently been established. The old 
weights and measures are commonly used: — th^ Castilian standards of the 
old Spanish system. The Quintal = loi. 4 pounds; the Arroba = 25.35 
pounds; the Fanega=i.5 bushels. 

URUGUAY. 

The French metric sjrstem was established in 1864. The old weights 
and measures are the same as those of the Argentine Confederation. The 
weights and measures of Brazil are in general use. 

PARAGUAY. 

The weights and measures are the same as the old ones of the Argentine 
Confederation. 

AUSTRALASIA. 

In New South Wales, Queensland, Victoria, South Australia, West 
Australia, Tasmania, and New Zealand, the legal weights and measures are 
the same as those of Great Britain. But the old British measures of 
capacity are also much used. 

In land measurement, a "section" is an area equal to 80 acres. 



MONEY. 



GREAT BRITAIN AND IRELAND. 



Coins. 



Material. 



Weight. 
Grains. 



j{d, farthing bronze. 



}id. halfpenny do. 

4 farthings i penny do. 

3//. threepenny piece silver. 

4i/. groat, or fourpenny piece do. 

6d. sixpence do. 

12 pence i shilling do. 

2 shillings i florin do. 

2j^x. I half-crown do. 

lox. I half-sovereign gold.. 

20s, I sovereign, or pound sterling do. 



• 43- 750 
. 87.500 

.145-833 
. 21.818 

. 29.091 

• 43.^36 
. 87.273 

.174.545 
.218.182 

. 61.6372 

.123.2745 



The bronze coins are made of an alloy of copper, tin, and zinc; the 
silver coins contain 92 J^ per cent, of fine silver, and Tj4 per cent of alloy; 
the gold coins, 91^ per cent, of fine gold, and 8^ per cent, of alloy. 

The Mint price of standard gold is ;£'3, ijs, ioJ4//. per ounce. 

One pound weight of silver is coined into 66 shillings. The intrinsic 
value of 22 shillings is equal to £1 sterling. 

The intrinsic vdue of 480 pence is equd to £1 sterling. 

FRANCE.— Money. 



Vxoo firanc . 
V50 franc 
Vao franc 



Copper, 

Coins. Weight. Value in English Money. 

Grammes. £ S, a, 

1 centime i o 

2 centimes 2 o 

5 centimes (j^«) 5 o 



'/xo franc 10 centimes (gr^jj^w)... 10 .0 



o 
o 
o 
o 






xo 



'A 

I 

2 

5 



franc, 
franc, 
franc. 



Silver, 

20 centimes i... 

50 centimes 2.5. 

100 centimes 5 



.0 
.0 

.0 



o 2 

o aM, 
o g}4 



francs 10 

francs 25 



more exactly 9.524//. 



.0 
.0 



I 
3 



7 



GERMANY, HANSE TOWNS. I9I 

GoU. 

Grammes. JZi ^' "• 

5 francs. 1*61290 o 3 11^ 

10 francs. 3*22580 o 7 ii^ 

20 francs (Nj^)oleon)... 6*45161 (99*56 grains)...o 15 10^ 

50 francs 16*12902 i 19 8^5 

100 francs 32*25805 3 19 4 4/,^ 

The English value is calculated at the rate of 25 francs 20 centimes to 
j£i. The bronze coins consist of an alloy of 95 parts of copper, 4 of tin, 
I of zinc. The standard fineness of the gold pieces, and of the silver 
5-franc pieces is 90 per cent., with 10 per cent of copper; of the other 
alver coins, 83.5 per cent.; and of the bronze coins, 95 per cent. 

GERMANY.— Money. 

The following system of currency was established throughout the German 

Empire in 1872: — 

English Value. 
s, d. 

I Pfennig = o .1175 

ID Pfennig i Groschen = o 1.175 

10 Groschen i Mark = o iij^ 

10 Marks (gold).. = 9 9^ 

20 Marks (gold)... = 19 7 

The 20-mark gold piece weighs 122.92 grains, and the standard fineness 
of the gold pieces is 90 per cent of gold. 

Before 1872, accounts were reckoned in the following currency in North 
Germany : — 

s, d, 

12 Pfennig i Silbergroschen = i i '/s 

30 Silbergroschen i Thaler = 30 

In South Germany: — 

4 Pfennig i Kreutzer = o ^ 

60 Kreutzers. i Florin = 18 

HANSE TOWNS.— Money. 

The monetary system is that of the German Empire. 
Hamburg. — ^According to the old monetary system, in which silver was 
the standard, 12 Pfennig =1 Schilling = ^ltd.\ and 16 Schillings =1 Mark 

= i3>i^. 
Bremen. — Old system: — 5 Schmaren= i Groot = "/aorfl; and 71 Groots = 

I Rix-dollar=3J. z^li^- The Rix-dollar, or Thaler, was a money of 

account 

Luhec. — ^The old system was the same as that of Hamburg, and, in 

addition, 3 Marks = i Thaler = 3 j. 4^/. 



192 MONEY. 

AUSTRIA.— Money. 

J. d, 

I Kreutzer (copper) o '/j 

4 Kreutzers (do.) o 4/^ 

10 Kreutzers (silver) o 2^ 

20 Kreutzers i^o.) o 4^^ 

^ Florin (do.) o sH' 

1 Florin (do.) i 11^ 

2 Florins (do.) 3 11}^ 

4 Florin piece (gold) 7 11 

8 Florin piece (do.) 15 10 

100 Kreutzers make i Florin. 

The 4-florin gold piece weighs 49.92 grains, and the standard of fineness 
is 90 per cent of gold. 

RUSSIA. — Money. 



I Copeck 

1 00 Copecks I Silver Rouble . 





s. 


d. 


= 





.38 


— 


3 


2 



The copper coins are pieces of ^, ^, i, 2, 3, 5 Copecks. The silver 
coins are pieces of 5, 10, 15, 20, 25 Copecks, the Half Rouble, and the 
Rouble; the gold coins are the Three-rouble piece, the Half Imperial of 
five Roubles, and the Imperial of 10 Roubles. The 5-rouble gold piece 
weighs 1 01 grains, and the standard of fineness is 91^ per cent of gold. 
Paper currency: — i, 3, 5, 10, 25, 50, 100 Roubles. 

HOLLAND.— Money. 

s, d, 

I Cent = o '/j 

100 Cents I Guilder or Florin = i 8 

BELGIUM.— Money. 
The monetary system is exactly the same as that of France. 

DENMARK.— Money. 

s, d, 

I Skilling = o .2745 

16 Skillings i Mark = o 4.392 

96 Skillings, or 6 Marks i Rigsdaler, or Daler = 22 7/ao 

SWEDEN. — Money. 

s, d, 

I Ore = o .133 

100 Ore I Riksdaler = i i^ 

NORWAY.— Money. 

s, d. 

I Skilling = o .444 

24 Skillingen i Ort or Mark = o lo^ 

5 Ort I Species-Daler = 4 $)i 



SWITZERLAND, SPAIN, ETC I93 

SWITZERLAND.— Money. 

The monetary system of Switzerland is the same as that of France. The 
Centime is called a Rappe. 

SPAIN. — Money. 

d. 

I Centimo = 95 

100 Centimos i Peseta = i franc, or 9)^ 

The bronze coins are pieces of i, 2, 5, and 10 centimos. The silver 
coins are pieces of 20 and 25 centimos, and i, 2, and 5 pesetas. The gold 
coins are pieces of 5, 10, 20, 25, 50, and 100 pesetas. The piece of 
5 pesetas is y, iij4^., English value. The 25 peseta piece is 19J. 9^^., 
English value. 

The old monetary system was based on the Real-Vellon, 2j^^. English 
value; it was the 20th part of the Silver Hard Dollar^ 4^. 2//. English value, 
and of the Gold Dollar or Coronilla. The Duro was identical with the 
American Dollar. 

PORTUGAL.— Money. 

The unit of account is the Rei, of which 185^ Reis make i penny; and 
4500 Reis make i sovereign. The Milreis is 1000 Reis, ^r. syid, English 
value. The Corda is the heaviest gold coin, of 10,000 Reis, JQ2, ^r. 5^//. 
English value, and weighs 17.735 grammes. 

ITALY. — Money. 

d, 

I Centime = .95 

100 Centimes i Lira = i franc, or 9^ 

Copper coins are pieces of i, 3, and 5 Centimes; silver coins, 20 and 
50 Centimes, and i, 2, and 5 Lire; gold coins, 5, lo, 20, 50, and 100 Lire. 
lliese coins are the same in weight and fineness as the coins of France. 

TURKEY. — Money. 

s, d, 

I Para = o 7,8.5 

40 Paras i Piastre = o 2.16 

100 Piastres i Medjidie, or Lira Turca = 18 o 

The Piastre is roughly taken equal to 2d. sterling. 

GREECE AND ;ONIAN ISLANDS.— MONEY. 

icx) Lepta i Drachma = i franc, or 9 J^//. 

The currency of Greece is the same as that of France. 

In the Ionian Islands, whilst they were under British protection (1830- 
1864), accounts were kept by some persons in Dollars, of 100 Oboli = 4f. 2//.; 
by others in Pounds, of 20 shillings, of 12 pence, Ionian currency; the 
Ionian Pound being equal to 20s. g,6d sterling. By other persons accounts 
were kept in Piastres of 40 Paras =^ 2 */^^, 

13 



194 MONEY. 

MALTA. — Money. 

I Grano 

20 Grani i Taro 

12 Tari I Scudo 

Or, 

60 Piccioli I Carlino 

9 Carlini i Taro 

12 Tari i Scudo 



s. 


d. 





v» 





I?^ 


I 


8 





.185 





x?^ 


I 


8 



British money is in general circulation. The Sovereign = 12 Scudi; the 
Shilling = 7 Tari 4 Gram. 

EGYPT. — Money. 

£ J. //. 

I Para = 00 .0615 

40 Paras i Piastre (Tariff) = o o 2.461 

100 Piastres i Egyptian Guinea = i o 6.84 

5 Egyptian Guineas... I Kees, or Purse = 52 10.2 

1000 Purses I Khuzneh, or Treasury = 5142 10 o 

97.22 Piastres i English Sovereign. 

The Egyptian guinea weighs 132 grains, and the standard of fineness is 
87^ per cent of gold. 
Two piastres (current) are equal to one piastre (tariff). 

MOROCCO.— Money. 

I Flue 

24 Flues I Blankeel = 

4 Blankeels i Ounce = 

10 Ounces i Mitkul = 

TUNIS. — Money. 

I Fel =0 3S/,88 

3 Fels I Karub = o 3S/^ 

16 Karubs i Piastre = o S ^/e 

ARABIA. — Money. 

s. d 
80 Caveers i Piastre or Mocha Dollar =35 

CAPE OF GOOD HOPE. — MoNEY. 

Public accounts are kept in English money; but private accounts are 
often kept in the old denominations, as foUows : — 

I Stiver = 

6 Stivers i Schilling = 

8 Schilling i Rix-doUar = 

The Guilder is equal to 6d. 



s. 


//. 





37/960 





3V40 





3-7 


3 


I 



s. 


d 





3/8 





^H 


I 


6 



INDIAN EMPIRE, CHINA, ETC 1 95 

I14DIAN EMPIRE.— Money. 

Throughout India, accounts are kept in the following moneys: — 

s, d. 

I Pie = o o^ nominal value. 

12 Pies I Anna. = o ij4 do. 

16 Annas i Rupee = 20 do. 

The intrinsic value of the Rupee is is. io}id.; it weighs 180 grains. 
The English Sovereign is equal to 10 Rupees 4 Annas. 

I Lac of Rupees = 100,000 rupees = ^10,000. 
I Crore of Rupees =100 lacs = ;;^i, 000,000. 

In Ceylon, the Rupee is divided into 100 Cents. 

The gold coin, Mohur, is equal to 15 rupees; it weighs 180 grains, and 
the standard fineness is 91.65 per cent of gold. 

CHINA.— Money. 

J. a. 

I Cash (Le) = o 7/,oo 

10 Cash I Candajreen (Fun) = o y/,© 

10 Candareens i Mace (Tsien) =0 7 

10 Mace iTael(Leang) = 5 10 

COCHIN-CHINA.— Money. . 

s. a, 

I Sapek, or Dong, or Cash = o «/i8 

60 Sapeks i Mas, or Mottien = o 3^ 

10 Mas. I Quan, or String. = 2 9^ 

PERSIA.— Money. . 

s, a. 

I Dinar = o '/a© 

50 Dinars i Shahi =0 ^ 

20 Shahis i Keran = o iij^ 

10 Kerans i Toman = 9 3J6 

JAPAN.— Money. 

10 Rin. I Sen = J^ 

100 Sen I Yen =42 

There are gold coins of the value of i, 2 and 5 yen, with a standard 
fineness of 90 per cent The 5-yen piece weighs 128.6 grains. The silver 
yen weighs 416 grains, with the same standard of fineness. 

JAVA. — Money. 
The money account of Java is the same as that of Holland. 

UNITED STATES OF AMERICA.— MoNEY. 

s. d, 

I Cent =0 }i 

10 Cents I Dime.... =05 

100 Cents I Dollar. =42 



196 MONEY. 

CANADA.— BRITISH NORTH AMERICA.— MoNEY. 

s. d. 

I Mil = o Vao sterling. 

10 Mils I Cent = o J^ do. 

100 Cents I Dollar = 4 i}( do. 

4 Dollars --= 20 o currency. 

Or, 

I Penny currency = o ^ sterling. 

12 Pence i Shilling do = o 94/^ do. 

20 Shillings i Pound do = 16 5^ do. 

The Dollar of Nova Scotia, New Brunswick, and Newfoundland, is equal 
to 4J. 2d. sterling. In the Bermudas, accounts are kept in sterling money. 

MEXICO.— Money. 

Accounts are kept in dollars of 100 cents. The dollar is equal to 4s, 2d. 
sterling. 

CENTRAL AMERICA AND WEST INDIES.— MoNEY. 

WEST INDIES (British). 

Accounts are kept in English money; and sometimes in dollars and 
cents. I dollar = 4r. 2d, 

CUBA. — Money. 

The moneys of various nations were in circulation before the current 
war (1875). But the principal silver currency was the 10 cent and 5 cent 
pieces of the United States. The gold currency consists of the Ounce, of 
the value of 16 dollars, }4 ounce, jounce, }i ounce. 

GUATEMALA, HONDURAS, COSTA RICA. 
The mone)rs of account are the same as those of Mexico. 

ST. DOMINGO. 

Accounts are kept in current dollars (called Gourde) and cents. The 
cent= Vsa^'i ^^d ^00 cents = i dollar = 3^//. 

SOUTH AMERICA. — MoNEY. 

COLOMBIA, VENEZUELA, ECUADOR. 

The moneys of account are, the Centavo= J^^.; and 100 Centavos = 
I Peso = 4J'. 2d, 

GUIANA. 

In British Guiana the dollar of 4s, 2d, is used, divided into 100 cents. 
In French Guiana, French money is used. In Dutch Guiana, the money of 
Holland is used. 



BRAZIL, PERU, ETC. 1 97 

BRAZIL. — Money. 

s. d. 

I Rei - o «Vioo 

looo Reis '. I Milreis - 2 3 

PERU. — Money. 

s. d, 
I Centesimo =0 .37 

1 00 Centesimos i Dollar, or Peso =31 

CHILI. — Money. 

J. d/ 
I Centavo =0 .45 

100 Centavos i Dollar, or Peso =39 

BOLIVIA. 

I Centena =0 .37 

100 Centenas i Dollar =31 

ARGENTINE CONFEDERATION. 

I Centesimo =0 .25 

100 Centesimos i Dollar, or Patercon =21 

URUGUAY. 

I Centime =0 oj4 

100 Centimes i Dollar =42 

PARAGUAY. 

I Centena =0 .37 

100 Centenas i Dollar =31 

AUSTRALASIA. 
Accounts are kept in pounds, shillings, and pence sterling. 



WEIGHT AND SPECIFIC GRAVITY. 



The specific gravity, or specific weight of a body, is the ratio which the 
weight of the body bears to the weight of another body of equal volume 
adopted as a standard for comparison of the weights of bodies. For solids 
and liquids, pure water at the mean temperature 62** F., is adopted as the 
standard body for comparative weight. For gases, dry air at 32° F., and 
under one atmosphere of pressure, or 14.7 lbs. per square inch, is the body 
with which they are compared. 

The specific gravity of bodies is found by weighing them in and out of 
water, according to the following rules. 

Rule i. — To find the specific gravity of a solid body heavier than water. 
Weigh it in pure water at 62® F., and divide its weight out of water by the 
loss of weight in the water. The quotient is the specific gravity. 

Note, — The loss of weight in water is the difference of the weight in air 
and the weight in water, and it is equal to the weight of the quantity of 
water displaced, which is equal in volume to the body. 

Rule 2. — To find the specific gravity of a solid body lighter than water. 
Load it so as to sink it in pure water at 62'' F., and weigh it and the load 
together, out of water, and in water; weigh the load separately in and out 
of water; deduct the loss of weight of the load singly from that of the 
combined body and load ; the remainder is the loss of weight of the body 
singly, by which its weight out of water is to be divided. The quotient is 
the specific gravity. 

Rule 3. — To find the specific gravity of a solid body which is soluble in 
water. Weigh it in a liquid in which it is not soluble; divide the weight 
out of the liquid by the loss of weight in the liquid, and multiply by the 
specific gravity of the liquid. The product is the specific gravity of the 
body. 

Rule 4. — To find the specific gravity of a liquid. Weigh a sblid body in 
the liquid and in water, as well as in the air, and divide the loss of weight 
in the liquid by the loss of weight in water. The quotient is the specific 
gravity. 

Rule 5. — To find the weight of a body when the specific gravity is given. 
Multiply the specific gravity by 

MULTIPLIES. WEIGHT OP 

62.355 (t^^ weight in pounds of a cubic foot of 

piure water at 62° F.) = i cubic foot, in lbs. 

1683.60 =1 cubic yard, in lbs. 

15.0 =1 „ incwts. 

.75 ^i „ intons. 



WEIGHT AND SPECIFIC GRAVITY. 1 99 

Note, — ^As one cubic foot of water at 62° F. weighs about 1000 ounces 
(exactly 997.68 ounces), the weight in ounces of a cubic foot of any other 
substance will represent, approximately, its specific gravity, supposing 
water =1000. 

If the last three places of figures be pointed off as decimals, the result 
will be the specific gravity approximately, water being = i. 

In France, the standard temperature for comparison of the density of 
bodies, and the determination of their specific gravities, is that of the 
maximum density of water, — about 4° C, or 39°.! F., for solid bodies; and 
32' F., or 0° C, for gases and vapours, under one atmosphere or .76 centi- 
metres of mercury. In practice, it is usual to adopt the cubic decimetre or 
litre as the unit of volume, since the cubic decimetre of distilled water, at 
4* C. weighs, by the definition, i kilogramme. Consequently the specific 
gravity of a body is expressed by the weight in kilogrammes of a cubic 
decimetre of that body. 

The densities of the metals vary greatly. Potassium and one or two 
others are lighter than water. Platinum is more than twenty times as 
heavy. Lead is over eleven times as heavy; and the majority of the useful 
metals are from seven to eight times as heavy as water. 

Stones for building or other purposes vary in weight within much 
narrower limits than metals. With one exception, they vary from basalt and 
granite, which are three times the weight of water, to volcanic scoriae which 
are lighter than water. The exception referred to is barytes, which is con- 
spicuously the heaviest stone, being 4j^ times as heavy as water. The 
sulphate of baryta is known as heavy spar. 

Amongst other solids, flint-glass has three times the weight of water; clay 
and sand, twice as much; coal averages one and a half times the weight of 
water; and coke from one to one and a half times. Camphor has about 
the same weight as water. 

Of the precious stones, zircon is the heaviest, having four and a half 
times the weight of water; garnet is four times as heavy, diamond three 
and a half times as heavy, and opal, the lightest of all, has just twice the 
weight of water. 

Peat varies in weight from one-fifth to a little more than the weight of 
water. 

The heaviest wood is that of the pomegranate, which has one and a third 
times the weight of water. English oak is nearly as heavy as water, and 
heart of oak is heavier; the densest teak has about the same weight as 
water; mahogany averages about three-fourths, elm over a half, pine from 
a half to three-fourths, and cork one-fourth of the weight of water. Of the 
colonial woods, the average of 22 woods of British Guiana weighs 74 per 
cent, of the weight of water; of 36 woods of Jamaica, 83 per cent.; and of 
18 woods of New South Wales, 96 per cent 

Wood-charcoal in powder averages one and a half times the weight of 
water; in pieces heaped, it averages only two-fifths. Gunpowder has about 
twice the weight of water. 

Of animal substances, pearls weigh heaviest, two and three-quarter times 
the weight of water; ivory and bone twice, and fat over nine-tenths the 
weight of water. 

Of vegetable substances, cotton weighs about twice as much as water; 
gutta-percha and caoutchouc nearly th^ §apa^ weight ^ waiter, 



200 WEIGHT AND SPECIFIC GRAVITY 

Mercury, the heaviest liquid at ordinary temperatures, has over thirteen 
and a half times the weight of water; and bromine nearly three times the 
weight The water of the Dead Sea is a fourth heavier, and ordinary sea- 
water two and a half per cent, heavier than water; whilst olive-oil is about 
one-tenth lighter, and pure alcohol and wood-spirit a fifth lighter than 
water. 

Turning to gaseous bodies, water at 62** F. has 772.4 times the weight of 
air at 32° F., under a pressure of one atmosphere; and the specific gravity 
of air at 32° F. is .001293, that of water at 62*^ F. being = i. Oxygen gas 
weighs a tenth more than air, gaseous steam weighs only five-eighths of air, 
and hydrogen, the most perfect type of gaseity, has only seven per cent, of 
the weight of air. Water has upwards of 11,000 times the weight of 
hydrogen. 

One pound of air at 62° F. has the same volume as a ton of quartz. 

The following Tables, Nos. 65 to 69, contain the weights and specific 
gravities of solids, liquids, and gases and vapours. The specific gravities 
have been derived from the works of Rankine, Ure, Wilson, Claudel, and 
Peclet, Delabfeche* and Playfair, Fowke, and others whose names are men- 
tioned in the body of the tables. Columns containing the bulks of 
bodies have been added to the tables. 

The specific gravity of alloys does not usually follow the ratios of those 
of their constituents; it is sometimes greater and sometimes less than the 
mean of these. Ure gives the specific gravities of some alloys of copper, tin, 
zinc, and lead, examined by Crookewitt. The following are the specific 
gravities of the alloys, as ascertained by Crookewitt; and, for the purpose 
of comparison, they are preceded by the specific gravities of the particular 
samples of the elementary metals employed. 

SPECIFIC GRAVITY. 

Copper 8.794 

Tin 7.305 

Zinc 6.860 

Lead ii-3S4 

Alloys: — Copper 2, tin 5 7«6S2 

Copper I, tin i 8.072 

Copper 2, tin i 8.512 

Copper 3, zinc 5 7.939 

Copper 3, zinc 2 8.224 

Copper 2, zinc i 8.392 

Copper 2, lead 3 10-753 

Copper I, lead i io-375 

Tin I, zinc 2 7.096 

Tin I, zinc I 7-ii5 

Tin 3, zinc i 7.235 

Tin I, lead 2 9*965 

Tin I, lead I 9*394 

Tin 2, lead I 9.025 

The following binary alloys have, on the one side, a density greater than 
the mean density of their constituents; and, on the other side, a density 
less than the mean density of the constituents. 



OF METALS AND ALLOYS. 



20 1 



Alloys liaving a density greater than the mean. 

Gold and zinc. 
Gold and tin. 
Gold and bismuth. 
Gold and antimony. 
Gold and cobalt 
Silver and zinc. 
Silver and lead. 
Silver and tin. 
Silver and bismuth. 
Silver and antimony. 
Copper and zinc. 
Copper and tin. 
Copper and palladium. 
Copper and bismuth. 
Lead and antimony. 
Platinum and molybdenum. 
Palladium and bismuth. 



Alloys having a density less than the mean. 

Gold and silver. 
Gold and iron. 
Gold and lead. 
Gold and copper. 
Gold and iridium. 
Gold and nickel. 
Silver and copper. 
Iron and bismuth. 
Iron and antimony. 
Iron and lead. 
Tin and lead. 
Tin and palladium. 
Tin and antimony. 
Nickel and arsenic. 
Zinc and antimony. 



202 



VOLUME, WEIGHT, AND SPECIFIC GRAVITY 



TABLE No. 65.— VOLUME, WEIGHT, AND SPECIFIC GRAVITY 

OF SOLID BODIES. 



FAMILIAR METALS. 

Platinum 

Gold 

Mercury, fluid 

Lead, milled sheet 

Do. wire : 

Silver 

Bismuth 

Copper, sheet 

Do. hammered 

Do. wire 

Bronze: — 84 copper, 16 tin, gun metal 

83 » 17 » « 

81 » 19 i> « 

79 „ 21 „ mill-bearmgs 

35 » 65 „ small bells 

21 » 79 V » 

15 „ 85 „ speculum metal... 

Nickel, hammered 

Do. cast 

Brass: — cast 

75 copper, 25 zinc, sheet 

66 „ 34 n yellow 

60 „ 40 „ Muntz's metal, ... 

Brass, wire 

Manganese 

Steel : — Least and greatest density 

Homogeneous metal 

Blistered steel 

Crucible steel 

Do. average '. 

Cast steel, 

Do. average 

Bessemer steel 

Do. average 

M ean for ordinary calculations 

Iron, wrought : — Least and greatest density... 

Common bar 

Puddled slab 

Various — Irons tested by Mr. Kirkaldy 

Do. average 

Common Tails 

Do. average 

Yorkshire iron bar 

Lowmoor plates, i|i to 3 ins. thick.... 

Beale's rolled iron 

Pure iron (exceptional), by electro- ) 

deposit (Dr. Percy) ) 

Mean, for ordinary calculations 



Weight of one 
cubic foot 



pounds. 



.. 1342 ... 
I2CO 
849 ... 
712 
704 ... 

655 
617 ... 

549 
.. 556 ... 

554 

534 ... 
528 

.. 520 ..• 

544 

503 ... 
461 

465 ... 

541 
516 ... 

505 
527 ... 

518 

511 

533 

499 ... 

435 to 493 

493 ... 
488 

..488 to 490... 

489 
489 to 489.5 

489.3 

..489 to 490... 
489.6 

489.6 ... 
466 to 487 

471 ... 

460.5 to 474 
..468 to 486... 

477 
..466 to 476... 

470 

484 ... 
487 
476 ... 



Specific Gfavity. 



Watcr= I. 



508 
480 



... 21.522 

19.245 . 
... 13.596 
II.418 
... 11.282 
10.505 
9.90 
8.805 
... 8.917 
8.880 
... 8.56 
8.46 
8.46 

8.73 
... 8.06 

7.39 
... 7.45 

8.67 
... 8.28 

8.10 

... 8.45 

8.30 

8.20 

8.548 
,.. 8.00 

7.729 to 7.904 

.. 7.904 
7.823 

7.825 to 7.859 

7.842 
7.844 to 7.851 

7.848 

7.844 to 7.857 

7.852 
.. 7.852 

7.47 to 7.808 

.. 7.55 

7.53 to 7.60 
.. 7.5 to 7.8 

7.65 

7.47 to 7.64 

7.54 
.. 7.758 

7.808 
.. 7.632 



8.140 

7.698 



OF SOLID BODIES. 



203 



Familiar Metals {continued). 

Iron, cast: — Least and greatest density 

White 

Gray ; 

Eglinton hot-blast, ist melting... 

2d do. . . . 
14th do. ... 

Rennie 

Mallett 

Mean, for ordinary calculations.. 

Tin 

Zinc, sheet 

Do. cast 

Antimony 

Aluminium, wrought 

Do. cast 

Magnesium 



OTHER METALS. 



Iridium 

Uranium 

Tungsten 

Thallium 

Palladium..... 

Rhodium 

Osmium 

Cadmium 

Molybdenum 
Ruthenium... 

Cobalt 

Tellurium 

Chromium.... 

Arsenic 

Titanium 

Strontium 

Glucinum 

Calcium 

Rubidiimi 

Sodium 

Potassium...., 
Lithium 



Weisht of one 
cubic fooL 



pounds. 

378.25 to 467.66 

468 

449 - 

435 

435 ... 
470 

435 to 444... 

442 

450 ... 
462 

449 ..- 
428 

418 

167 

160 

108.5 



1 165.0 

1 147.0 

1097.0 

742.6 

735.8 
660.9 
623.6 

542.5 

537.5 
536.2 

530.0 

381.0 

374.1 
361.5 

330.5 
158.4 

131.0 
98.5 

94.8 
60.5 

53-6 
37.0 



Specific Gravity. 



Water = i. 

6.900 to 7.500 

7.50 

7.20 

6.969 

6.970 

7.530 

6.977 to 7. 1 13 

7.094 
... 7.217 

7.409 
7.20 

6.86 
6.71 
2.67 
2.56 
1.74 



PRECIOUS STONES. 



Zircon 

Garnet 

Malachite 

Sapphire 

Emerald 

Do. Aqua marine.. 

Amethyst 

Ruby 

Diamond. 



Specific Gravity. 
.. 4.50 

3.60 to 4.20 

.. 4.01 

3.98 

.. 3.95 

2.73 
.. 3.92 

3.95 

3.50 to 3.53 



Diamond, Pure 

Boart 

Topaz 

Tourmaline 

Lapis lazuli 

Turquoise 

Jasper, Onyx, Agate.... 

Beryl.: 

Opal 



18.68 

18.40 

17.60 

11.91 

11.80 

1 0.60 

10.00 

8.70 

8.62 

8.60 

8.50 

6.1 1 

6.00 

5.80 

5.30 
2.54 

2.10 
1.58 
1.52 
0.97 
0.86 

0.59 



Specific Gravity. 
... 3.52 

3.50 
... 3.50 

3.07 
... 2.96 

2.84 

2.6 to 2.7 
2.68 



2.09 



204 



VOLUME, WEIGHT, AND SPECIFIC GRAVITY 



STONES. 

Specular, or red iron ore 

Magnetic iron ore 

Brown iron ore 

Spathic iron ore , 

Clydesdale iron ores , 

Barytes 

Basalt 

Mica 

Limestone, Magnesian 

Do. Carboniferous 

Marble: — Paros 

African 



Pyrenean , 

Egyptian, green 

French 

Florentine, Sienna 

Trap, touchstone 

Granite, Sienite, gneiss 

Do. Gray 

Porphyry 

Alabaster, Calcareous 

Do. Gypseous 

Chalk, Air-dried 

Slate 

Serpentine 

Potter's Stone 

Schist, Slate 

Do. Rough 

Lava, Vesuvian 

Talc, Steatite 

Rock Crystal 

Quartz 

Do. Crystalline 

Do. for paving 

Do. porous, for millstones 

Do. flaky, for do 

Flint 

Felspar 

Gypsum 

Lias 

Graphite 

Sandstone 

Tufa, volcanic 

Scoria, do 



Cubic feet to one 
ton, solid. 



cubic feet. 

6.84 • * 
7.05 
9.16 .. 

9.38 
.. 11.76 .. 
8.07 

14-7 to 12.0 
14.0 to 12.3 
,. 12.6 .. 

13.3 
.. 12.7 .. 

12.8 
.. 13.2 .. 

13.2 
.. 13.2 .. 

13.5 
.. 13.6 .. 

14.3 
.. 13.2 

15.2 to 12.1 

12.8 to 11.8 

13-5 to 13.1 

.. 13.0 .. 

15.6 

14.9 to 14. 1 

13.8 to 12.6 
.. 12.8 .. 

12.8 
.. 12.8 

19.9 to 12.9 
21.0 to 12.8 

13.3 
13.6 

13.8 to 13.3 

13.6 

.. 14.4 .. 
28.5 

I4.I 

13-7 
.. 13.8 .. 

15.6 

16.0 to 14.7 

16.3 

17.3 to 14.3 
29,7 to 26.1 

.. 43.3 •. 



Weight of one cubic 
foot, solid. 



pounds. 

••• 3274 

317.6 
... 244.6 .. 

238.8 

190.5 .. 
277.5 

152.8 to 187.1 
160.3 to 182.7 

... 178.3 .. 
168.0 

... I77.I .. 
174.6 
170.2 .. 
170.2 

169.6 .. 
166.5 
165.2 .. 

157.1 
169.6 .. 

147.1 to 184.6 
174.6 to 190.8 

166.5 to 171.5 
... I72.I 

144.0 

...150 to 159.. 
162.1 to 177.7 

... 175.2 .. 
174.6 

... 174.6 .. 
112.8 to 173.3 

106.6 to 175.2 

168.4 

165.2 

162.8 to 169.0 

165.2 

... 155.9 .. 
78.6 

... 159.0 .. 

164.0 
... I62.I .. 

1434 

140.3 to 152.8 

137.2 

129.7 to 1 57. 1 
75.4 to 86.0 

... 51.7 .. 



Specific Gravity. 



• B I •  • 



Water = I. 

5.251 

5.094 
... 3.922 

3.829 
3.055103.380 

4-45 

2.45 to 3.00 

2.57 to 2.93 

2.86 

2.69 
. .. 2.04 

2.80 
... 2.73 

2.73 

• •  i 

2.67 

2.65 

2.52 

... 2.72 

2.36 to 2.96 

2.80 to 3.06 
2.67 to 2.75 

... 2.76 
2.31 

2.46 to 2.55 

2.60 to 2.85 
... 2.81 

2.80 

2.80 

1. 81 to 2.78 
1.71 to 2.81 

2.70 
2.65 

2.61 to 2.71 
2.65 

... 2.50 
1.26 

... 2.55 

2.63 

2.60 

2.30 
2.25 to 245 

2.20 
2.08 to 2.52 

I.2I to 1.38 
•83 



1 



\ 



OF SOLID BODIES. 



201; 



SUNDRY MINERAL 
SUBSTANCES.' 

Glass:— Flint 

Green 

Plate 

Crown 

St. Gobain 

Common, with base of potash 

Fine, do. do. 

Common, with base of soda... 

Fine, do. do. ... 

Soluble 

Porcelain :— China 

Sevres 

Portland Cement 

Concrete : — 

P. cement i, and shingle 10 
P. cement, rubble, and sand 

P. cement i, and sand 2 

Roman cement i,and sand 2 

Mortar. 

Brick 

Brickwork 

Masonry, Rubble 

MarL 

Do. very tough 

Potash 

Sulphur 

TUes 

Rock Salt 

Conunon Salt, as a solid 

Clay 

Sand, pure 

earthy 

Earth :— Potter's 

Argillaceous 

Light vegetable 

Mud 

Materials in the bed of the Clyde :- 

Fine sand and a few pebbles, 

laid in a box, loose, not 

pressed, nearly dry 

Pressed 

Mud at Whiteinch, dry, and ] 
firmly packed, containing > 

very fine sand and mica ) 

Wet mud, rather compact and 

firm,well pressed into the box 

Wet, fine, sharp gravel, well 

pressed 

Wet, running mud 

Sharp dry sand deposit, in ^ 

harbour ) 

Port-Glasgow bank (sand), wet, ) 
pressed into a box \ 



] 



Cubic feet to one 
ton, solid. 



cubic feet. 



Weight of one cubic 
foot, solid. 



28.7 to 23.8 

.. 16.1 

16.6 to 16.0 

.. 17.6 .. 
18.7 
20.6 

1 8. 1 to 16.0 
20.4 to 19.5 
19.4 to 15.6 
22.4 to 18.9 

.. 17.1 
18.0 
.. 18.0 .. 
17. 1 to 15.9 
.. 18.7 .. 

18.7 
.. 18.9 .. 

21. 1 
.. 18.9 .. 

22.4 
.. 25.7 .. 

22. 



pounds. 
187.0 
168.4 
168.4 

155-9 
155.3 
153-4 
152.8 

152.8 
1 52. 1 

77.9 
148.4 

139-7 
78 to 94 



26 
24 

23 

19 

18 
18.1 

24-3 

18.6 



.. 139 
135 to 140 

127 

120 

109 

124.7 to 135.3 
..no to 115. 
1 1 5.3 to 143.4 
99.8 to 1 18.5 

146 
.. 131 

124-7 
124.7 •• 
131 to 140.7 

.. II9.7 .. 
1 19.7 

.. II8.5 .. 

106.0 
.. II8.5 .. 

99.8 
87.3 .. 

1 01. 6 



Specific Gravity. 



87 
92 

97 

115 
124 

122X 
92 

120.5 



Water = i. 
3.00 
2.70 
2.70 
2.50 
2.49 
2.46 
2.45 
2.45 
2.44 
1.25 
2.38 
2.24 

1.25 to 1. 51 

.. 2.23 

2.17 to 2.25 

.. 2.04 

1.92 

.. 1.75 

2.00 to 2.17 
1.76 to 1.84 
1.85 to 2.30 
1.60 to 1.90 

2.34 

2.10 
2.00 
2.00 

2.IOOtO 2.257 
1.92 
1.92 
1.90 
1.70 
1.90 
1.60 
1.40 
1.63 



... 1.39 
1.48 

... 1.56 

... 1.95 

1.99 

1.97 

... 1.48 

... 1.93 



206 



VOLUME, WEIGHT, AND SPECIFIC GRAVITY 



Mineral Substances {continued). 

Materials in the bed of the Qyde ; — 
Sand opposite Erskine House, ) 

wet, pressed J 

Alluvial earth, pressed 

Do. do. loose 

Plaster: — 24 hours after using 

2 months after using ... 
Coal, Anthracite (see Sect. Coal) 
Bituminous do. do. 
Boghead (cannel) do. do. 

Coke 

Phosphorus 

Alum 

Camphor 

Meltmg Ice 



Cubic feet to one 
ton, solid. 



cubic feet. 



19.3 



24 

.. 33 
22.6 

.. 25.7 .. 

26.2 to 22.6 

30 to 28.1 

30 
39 to 21.6 
20.3 
20.9 .. 

36.3 

•• 39 



Weight of one cubic 
foot, solid. 



pounds. 



116 



93 

67 

99.2 

.. 87.3 .. 

85.4 to 99.1 

74.8 to 81.7 

74.8 
57.4 to 103.5 

1 10.4 

.. 107.2 .. 

61.7 

.. 57.5 .. 



Specific Gravity. 



Waters X. 

.. 1.86 

1.49 
.. 1.08 

1.59 
1.40 

1.37 to 1.59 

1.20 to 1. 31 

1.20 
.92 to 1.66 

1.77 
.. 1.72 

•99 
.922 



• • • 



COALS. 

Delabeche and Playfair,) 

Welsh: — ^Anthracite 

Porth Mawr (highest) 

Llynvi (one of the lowest) 

Average of 37 samples 

Newcastle: — Hedley's Hartley (highest) ... 
Original Hartley (one of the lowest) 

Average of 18 samples 

Derbyshire and Yorkshire: — Elsecar 

Butterley 

Stavely 

Loscoe, soft 

Average of 7 samples 
Lancashire: — Laffack Bushy Park (highest) 

Cannel, Wigan (lowest) 

Average of 28 samples 

Scotch : — Grangemouth (h ighest) 

Wallsend Elgin 

Average of 8 samples 

Irish : — Slievardagh Anthracite 

Warlich's artificial fuel 



Cubic feet 
in a ton. 



Heaped. 



cubic feet. 

38.4 
42.0 

42.0 

42.7 

431 
45.6 

45-3 

47.4 

47-3 

44.9 
48.8 

47.4 
42.6 
46.4 

45.2 
40.1 
41.0 
42.0 

35.7 
32.4 



Weight of one 
cubic foot. 



SoUd. 



pounds. 

85.4 
86.7 
80.3 
82.3 
81.8 
78.0 

78.3 
80.8 

79.8 

79.8 

79.6 

79.6 

84.1 

76.8 

79-4 
80.5 

74.8 
78.6 
99.6 
72.2 



Heaped. 



pounds. 
58.3 

53-3 

53.3 

53.1 
52.0 

49.1 

49.8 
47.2 

47.4 

49-9 

45.9 

45-9 
52.6 

48.3 

49.7 
54.3 
54.6 

50.0 
62.8 
69.6 



Specific 
Gravity. 



Water = i. 

.37 

.39 
.28 

•315 
•31 

.256 
.296 
.28 

.27 
.285 

.292 

•35 

.23 

.273 
.29 

.20 

.259 

.59 
•15 



OF SOLID BODIES. 



207 



PEAT. 

{Dr, Sullivan,) 

Irish peat (comprising an 
average amount of water 
from 20 to 25 per cent) : — 

Lightest upper moss peat ... 

Average lignt moss peat 

Average brown peat 

Compact black peat 

Mean of five samples 

{Another observation,) 
Average upper brown peat .. 
Moderately compact lower ) 

brown turf ) 

Mean of two classes 

Condensed peat 

{Kane and Sullivan,) 
Excessively light, spongy ) 

surface peat ) 

Light surface peat 

Rather dense peat 

Very dense dark brown peat 
Very dense blackish brown ) 

compact peat J 

Exceedingly dense jet black \ 

^P«at ( 

Exceedingly dense, dark, \ 

blackish brown peat \ 

(/CarmarscA.) 

Turfy peat, Hanover 

Fibrous peat, do 

Earthy peat, do 

Pitchy peat, do 



Cubic feet per 
ton, stalked. 



cubic feet. 



...369.60. 

254.20 
...147.00. 

131.28 
... 99.36. 

200.29 



. . . I oo.O . . . 



155-5 



...141.75. 
5 1.2 to 40.0 



Weight of one 
cubic foot, 

stalked. 



pounds. 



6.06 
8.81 

15-13 
17.06 

22.54 

11.18 



.. 11.92 

14.40 

,. 15.80 

43.75 to 56.8 



Weiffht of one 

cubic foot, 

solid. 



pounds. 



62.5 to 81. 1 

13.7 to 21.0 

20.9 to 25.3 
29.7 to 41.7 
40.5 to 44.5 

45.1 to 61.3 

53.2 to 61.8 

. . . 66.0 . . . 



6.9 to 16.2 
1 5.0 to 41.8 

2 5.6 to 56.1 

38.7 to 64.2 



Specific 
Gravity. 



Water = i. 



i.o to 1.3 



.2 19 to .337 

.335 to .405 
.476 to .669 

.65010.713 
.72410.983 

.72510.991 

... 1.058 

.II to .26 
.24 to .67 
.41 to .90 
.62 to 1.03 



FUEL IN FRANCE. 

{ClaudeL) 

Pure Graphite. 

Anthracite 

Rich coaly with ^ long flame 

Dry coal, with a long flame 

Rich and hard coal 

Smithy coaL 

Lignite 

Do. bituminous 

Do. imperfect 

«Jayet»...r! 

Bitumen, red 

Do. black 

Do. brown 

Asphalte 



Weisht of 
one cubic foot 



pounds. 



.. 145-3 •• 

83.5 to 91.0 

79.8 to 84.8 

84.8 
.. 82.3 .. 

79.8 to 81. 1 

77.9 to 84.2 
72.3 to 74.8 

68.6 to 74.2 
81.7 

.. 72.3 .. 

66.7 
... 51.7 .. 

66.1 



Specific Gravity. 



Water = x. 



.. 2.33 

1.34 to 1.46 
1.28 to 1.36 

1.36 
.. 1.32 

1.28 to 1.30 
1.25 to, 1.35 
1. 16 to 1.20 
1. 10 to 1. 19 

.. 1. 16 

1.07 
.. 0.83 

1.06 



2o8 



VOLUME, WEIGHT, AND SPECIFIC GRAVITY 



WOODS. 



Pomegranate 

Boxwood 

Do. of Holland 

Do. of France 

Lignum vitae 

Ebony 

Do. Green 

Do. Black 

Oak, Heart of. 

Do. English 

Do. European 

Do. American, Red 

Lancewood 

Rosewood 

Satin-wood 

Walnut, Green 

Do. Brown 

Laburnum 

Hawthorn 

Mulberry 

Plum-tree 

Teak, African » 

Mahogany, Spanish 

Do. St. Domingo 

Do. Cuba 

Do. Honduras 

Beech 

Do. with 20 per cent, moisture. 

Do. cut one year 

Ash.... 



Weight of one 
cubic foot. 



Do. with 20 per cent, moisture 

Acacia 

Do. with 20 per cent, moisture. 

Holly 

Hornbeam 

Yew 



Birch 

Elm 

Do. Green 

Do. with 20 per cent, moisture 

Yoke-Elm do. do 

Rock-Elm 

Fir, Norway pine 

Do. Red pine 

Do. Spruce 

Do. Larch 

Do. White pine, English 

Do. do. Scotch 

Do. do. do. 20 per cent, moisture.. 

Do. Yellow pine 

Do. do. American 

American Pine-wood, in cord (heaped) 

Apple-tree 



pounds. 
.. 84.2 .. 

64.8 

.. 82.3 .. 

56.7 

40.5 to 82.9 

70.5 

.. 75.5 .. 

74.2 

.. 730 .. 
58.0 

43.0 to 61.7 

54.2 

41.8 to 63.0 
64.2 

59-9 •• 

57.4 
42.4 .. 

57-4 
56.7 .. 

55.5 
54.2 .. 



Specific Gravity. 



53.0 .. 
46.8 

34-9 •• 

, 34.9 

46.8 to 53.0 

51.1 
... 41.2 .. 

52.4 
... 43.7 .. 

51.1 
... 44-9 •• 

47.5 

... 47.5 •• 

46.1 to 50.5 

44.9 to 46. 1 

34-3 
... 47.5 .. 

44-9 
... 47.5 .. 

50.0 

4^* ' 
29.9 to 43.7 

29.9 to 43.7 

31.18 to 39.9 

•.. 34.3 •• 
34.3 

30.0 .. 

41.2 

... 28.7 .. 

21 

... 45.5 .. 



•• 1-35 
1.04 

1.32 

0.91 

.65 to 1.33 

1. 13 
1.21 

1.19 

.. 1.17 

0-93 
.69 to .99 

.87 

.67 to 1. 01 

1.03 

.. 0.96 

0.92 

.. 0.68 

0.92 

.. 0.91 

0.89 

.. 0.87 

.98 

.. 0.85 

0.75 

.. 0.56 

0.56 
0.75 to 0.85 

0.82 

0.66 

a84 
.. 0.70 

0.82 
.. 0.72 

0.76 
.. 0.76 
0.74 to 0.81 
0.72 to 0.74 

0-55 
.. 0.76 

0.72 

0.76 

0.80 
.. 0.74 
0.48 to 0.70 
0.48 to 0.70 
0.50 to 0.64 

.. 0.55 

0.53 
.. 0.49 

0.66 
.. 0.46 

0.34 
.. 0.73 



OF SOLID BODIES. 



209 



1 



Pear-tree....: 

Orange-tree % 

Olive-tree 

Maple 

Do. 20 per cent, moisture 

Service-tree. 

Cypress, cut one year 

Plane-tree 

Vine- tree 

Aspen-tree., 

.\lder-tree 

Do. 20 per cent, moisture 

Sycamore 

Cedar of Lebanon 

Bamboo 

Poplar. 

Do. White 

Do. 20 per cent, moisture 

Willow 

Cork. 

Elder pith 

INDIAN WOODS. 

{Berkley^ 

Northern Teak 

Southern Teak 

Jungle Teak 

Blackwood 

Khair 

Enroul 

Red Eyne 

Bibla 

Peon 

Kullum 

Hedoo. 

COLONIAL WOODS. 

Jamaica:— 

Black heart ebony 

Lignum vitae 

Small leaf. 

Neesberry bullet-tree 

Red bully-tree 

Iron wood 

Sweet wood 

Fustic 

Satin candlewood , 

Bastard cabbage bark 

White dogwood 

Black do 

Gynip 



Weiffht of one 
cubic fooL 




pounds. 

45-5 

44-3 •• 
42.4 

40.5 .. 

41.8 

41.8 .. 
41.2 
40.5 .. 

37-4 
374 .. 
34.9 
37.4 .. 
36.8 

30.6 to 35.5 
19.5 to 24.9 
.. 24.3 .. 
20.0 to 31.8 

29.9 .. 

30.6 

15.0 .. 

4.74 



55 

48 

41 
56 

11 

63 
68 

56 

39 
41 
39 



40.5 



74.2 .. 
to 73.0 

73.0 .. 
65.5 
62.36 .. 
61.7 

60.5 .. 
60.5 

59-9 •• 
58.6 

58.6 .. 
58.0 
58.0 .. 



Specific Gravity. 



073 
.. 0.71 

0.68 
.. 0.65 

0.67 
.. 0.67 

0.66 
.. 0.65 

0.60 

0.60 

0.56 

0.60 

0.59 

0.49 to 0.57 
0.31 to 0.40 

•• 0.39 

0.32 to 0.51 
.. 0.48 

0.49 

.. 0.24 
0.076 



0.882 

0.770 

0.658 
0.898 

1. 171 
I.0I4 
I.09I 

0.898 

0.625 

0.658 

0.625 



1. 19 
0.65 to 1. 17 

.. 1. 17 

1.05 

I.OO 

0.99 

.. 0.97 

0.97 

.. 0.96 

0.94 
.. 0.94 

0-93 
.. 0.93 



14 



210 



VOLUME, WEIGHT, AND SPECIFIC GRAVITY 



Colonial Woods {continual). 

Jamaica {continued): — 

Wild mahogany 

Cashaw 

Wild orange 

Sweet do 

Bullet-tree (bastard) 

Tamarind 

Do. wild 

Prune 

Yellow Sanders 

Beech 

French Oak 

Broad Leaf 

Fiddle Wood 

Prickle Yellow 

Boxwood 

Locust-tree 

Lancewood 

Green Mahogany 

Yacca. 

Cedar 

Calabash 

Bitter Wood 

Blue Mahoe 

Average of 36 woods of Jamaica 

New South Wales:— 

Box of Ilwarra 

Do. Bastard 

Do. True, of Camden 

Mountain Ash 

Kakaralli 

Iron Bark 

Do. broad-leaved 

Woolly Butt 

Black Do 

Water Gum 

Blue Do 

Cog Wood 

Mahogany 

Do. swamp 

Gray Gum 

Stringy Bark 

Hickory 

Forest Swamp Oak 

Mean of 18 woods of New South Wales.. 
British Guiana :-- 

Sipiri, or Greenhcart 

Wallaba 

Brown Ebony 

Letter Wood 

Cuamara or Tonka 

Monkey Pot 

Mora 



Weiffht of one 
cubic foot. 



pounds. 



.. 57.4 .. 

574 
53.0 to 56.7 

49-3 
.. 56.1 .. 

54.2 

46.8 
. 53.6 
.. 53-6 .. 

52.4 

48.0 
.. 44.3 .. 

43-0 
.. 43.0 .. 

42.4 

.. 42*4 *• 
41.2 

.. 39-3 .• 
36.2 

.. 34-9 •• 

34.3 

.. 337 .. 
52.1 



73-0 
69.8 

60.5 

69.2 

68.6 

64.2 

63.6 

63.0 

55.5 
63.6 

52.4 

59-9 
59.2 

53.6 
58.0 

53.6 
46.8 
41.2 

59-9 



65.5 to 68.0 

64.8 

64*2 

62.36 
.. 61.7 .. 

58.6 



57.4 



Spedfic Gtavity. 



0.92 
0.92 

.85 to 0.91 

0.79 
0.90 
0.87 
0.75 
0.86 
0.86 
0.84 

0.77 
0.77 

0.71 

0.69 

0.69 

0.68 

0.68 

0.66 

0.63 

0.58 

0.56 

0.55 
0,54 
0.835 

. I.I7 
1. 12 
0.97 
I.II 
1. 10 
1.03 
1.02 
l.OI 
0.89 
1. 00 
0.84 
0.96 
0.95 
0.86 

0.93 

a86 

0.75 
0.66 

0.96 



1.05 to 1.09 

1.04 

1.03 

1. 00 
.. 0.99 

0.94 
.. 0.92 



J 



OF SOLID BODIES. 



211 



Colonial Woods {continued), 

British Guiana {continued) :— 

Ducaballi 

Cabacalli 

Kaiecri-balli 

Sirabuliballi 

Buhuradda 

Buckati 

Houbaballi 

Baracara. 

White Cedar 

Locust-tree. 

Cartan 

Purple Heart 

Bartaballi 

Crabwood : 

SilverbaJli 

Mean of 22 woods of British Guiana. 



' WiUow 
Oak 



WOOD-CHARCOAL (as powder). 
{Ciaudel) 



Alder. 

Lime-tree 

Poplar 

Average of 5 charcoals, 



WOOD-CHARCOAL (in small pieces, heaped). 



Walnut 

Ash 

Beech 

Voke-elm. 

Apple-tree 

White Oak..... 
^erry-tree.... 

Birch. 

EJm 

Yellow Pine.. 
Chestnut-tree. 

Poplar , 

Cedar 



{ClaudeL) 



Average of 13 charcoals 
Gunpowder 



WOOD-CHARCOAL (as made, heaped). 



Oak and Beech.. 

Birch 

Pine. 



Average. 



Weight of one 
cubic foot. 



pounds. 

56.7 

55.5 
54.2 

52.4 

50.5 

50.5 

50.5 

50.5 

48.0 

44.3 

437 
42.4 

39-9 

37.4 

34-3 
46.1 



96.7 

95-4 
92.9 

91.0 
90.4 
93.5 



39-3 

34-3 

32.5 
28.7 

28.7 

26.2 

25.6 

22.5 

22.5 

20.6 



17.5 ... 
15.6 

15.0 ... 

25.3 
i09.itoii4.7 



15 to 15.6 

137 to 14.3 

12.5 to 13.1 

14 



Specific Gravity. 



0.91 
0.89 
0.87 
0.84 
0.81 
0.81 
0.81 
0.81 
0.77 
0.71 
0.70 
0.68 
0.64 
0.60 

0.55 
0.74 



1-55 

1.53 
1.49 

1.46 

145 
1.50 



.. 0.63 

0.55 
.. 0.52 

0.46 
.. 0.46 

0.42 
.. 0.41 

0.36 
.. 0.36 

.. 0.28 

0.25 
.. 0.24 

0.405 
1.75 to 1.84 



0.24 to 0.25 
0.22 to 0.23 
0.20 to 0.21 
,.. 0.225 



212 



WEIGHT AND VOLUME OF 



ANIMAL SUBSTANCES. 

{Claudel) 

Pearls 

Coral 

Ivory 

Bone 

Wool 

Tendon 

Cartilage 

Crystalline humour 

Human body 

Nerve 

Wax 

White of whalebone 

Butter 

Pork fat 

Mutton fat 

Animal charcoal, in heaps 



VEGETABLE SUBSTANCES. 

{Claudel) 

Cotton 

Flax 

Starch 

Fecula 

Gum — Myrrh 

Do. Dragon 

Do. Dragon's blood 

Do. Sandarac 

Do. Mastic 

Resin — Jalap 

Do. Guayacum 

Do. Benzoin 

Do. Colophany 

Amber, Opaque 

Do. Transparent 

Gutta-percha 

Caoutchouc 

Grain, Wheat, heaped 

Do. Barley, do 

Do. Oats, do 



Weight of one 
cubic foot. 



pounds. 

... 169.6 ... 

167.7 

... 1 19.7 ... 
1 1 2.2 to 124.7 

... 100.4 ... 

69.8 

68.0 ... 

66.7 

64,9 
... 59.9 ... 

58.7 
... 58.7 ... 

58.7 



574 
50 to 52 



121.6 

1 1 1.6 

95-4 

93-5 
84.8 

82.3 

74.8 
68.0 

66.7 

76.1 

74-8 

68.0 

66.7 

68.0 

67.3 

60.5 

58.0 

46.7 

36.6 

31.2 



Specific Gravity. 



.. 2.72 
2.69 
1.92 

1.80 to 2.00 

I.61 
1. 12 



1.09 
1.08 
.1.07 
1.04 
0.96 
0.94 
0.94 

0.94 
0.92 



0.80 to 0.83 



O, 

o, 
o, 
o 
o 



95 
79 
53 
50 
36 

32 
20 

09 

07 

22 

20 
09 

07 
09 

08 
97 
93 
75 
59 
50 



VARIOUS SUBSTANCES. 



213 



TABLE No. 66.— WEIGHT AND VOLUME OF VARIOUS 

SUBSTANCES. {Tredgold,) 



SUBSTANCE. 



Lead (cast in pigs) 

Iron (cast in pigs) 

Limestone or marble (in blocks) 
Granite (Aberdeen, in blocks) ... 

Granite (Cornish, in blocks) 

Sandstone (in blocks) 

Portland stone (in blocks) 

Potter's clay 

Loam or strong soil 

Bath stone (in blocks) 

GraveL 

Sand. 

Bricks (common stocks, dry) 

Culm 

Water (river) 

Splint coal 

Oak (seasoned) 

Coal (Newcastle caking) 

Wheat 

Barley 

Red fir 

Hay (compact, old) 



Cubic feet per 
ton, in bulK. 



cubic feet. 

4 
6.25 

13 
13.5 
14 
16 

17 

17 
18 

18 

21 

23.5 
24 

36 

36 

39-5 

43 

45 

47 

59 

59 
280 



Weight of one 
cubic toot, in bulk, 



lbs. 
567 
360 
172 
166 
164 
141 

132 
130 
126 

123.5 
109 

95 
93 
63 
62.5 

57 
52 

50 
48 

38 

38 
8 



TABLE No. 67.— WEIGHT AND VOLUME OF GOODS CARRIED ON 
THE BOMBAY, BARODA, AND CENTRAL INDIA RAILWAY. 



By Colonel J. P. Kennedy, Consulting Engineer of the Railway. 



No. of 
kbuL 



I 

2 

3 
4 

I 

7 



Class I. 



CLASSIFICATION OF GOODS CONVEYED. 



Unpressed cotton . . . . 

Furniture 

Half-pressed cotton.. 

Cotton seeds 

Wool 

Fruit and vegetables. 
Eggs.... 

Averages 






Cubic feet 
per ton. 


Weight per ! 
cubic foot. 


cubic feet 
... 224 ... 

200 
... 186 ... 

186" 
... 140 ... 

100 
... 90... 


lbs. 
... 10 ... 

II 
... 12 ... 

12 
... 16 ... 

22 
... 25 ... 


...174... 


... 13 ... 



Cubic feet 
per ton, 
in bulk 

(estimated). 



cubic feet. 
.. 280 
250 

..233 

,..175 

125 

... 113 

... 217 



214 



WEIGHT AND VOLUME OF GOODS. 



Goods conveyed over the Indian Railway {continued). 



No. of 
kind. 



8 
9 

lO 

II 

12 

M 

17 
i8 

19 

20 
21 
22 



Class 2. 



classification of goods conveyed. 



23 ... 
24 

25 ... 
26 
27 ... 

28 

29 ... 

30 

31 .- 

32 

33 - 
34 

35 ..• 
36 



Class 3. 



37 ..• 

38 

39 •• 
40 

41 ... 

42 
43 ... 



Qass 4. 



Grass 

Sundries 

Bagging .,., 

Commissariat stores 

Full-pressed cotton 

Flax and hemp 

Groceries 

Grains and seed 

Twist ^ 

Sugar 

Soap 

Firewood 

Salt 

Lime 

Dry Fruits 

Averages 

Jagree (Molasses) 

Kupas (Seed cotton) 

Mowra (flowers which produce spirit) 

Timber 

Ghee (clarified butter) 

Oil 

Piece goods 

Rape 

Beer and Spirits 

Coal 

Paper 

Tobacco 

Opium 

Machinery 

Averages 

Cutlery 

Potash 

Sand 

Colour 

Bricks 

Stone 

Metal 

Averages 

Averages of all classes 



Cubic feet 
per ton. 



culnc feet. 



80 
80 
70 
70 
70 
70 
60 
60 
60 

56 

5> 
51 
50 



Weight per 
cubic foot. 



60 



45 

45 

45 

45 
40 

40 

40 

40 

36 
28 
28 
28 
26 

25 



... 41 



20 
20 
20 
18 
17 
15 
5 



... II 



. . . 64.4 ' 



lbs. 

28 

28 

32 
32 

32 
32 

37 

37 

37 
40 

40 

40 

44 
44 
45 



37 



50 
50 
50 

^A 

5^ 

5^ 

5^ 
56 

62 
80 
80 
80 
86 
90 



54 



112 
112 
112 
124 
132 
148 

443 



354 



Cubic feet 
per ton, 
m bulk 

(estimated). 



203 ... 



cubic feet. 
...ICX> 
ICO 
... 87 

87 
... 87 

87 

... 75 

75 
... 75 

70 
... 70 

70 
... 64 

64 
... 63 



... 75 



56 

56 
50 
50 
50 
50 
45 
35 
35 
35 
33 
31 



51 



25 

25 
20 

22 

21 

19 
6X 



... 14 



80 



J^oU, — The last column has been added bv the author; the quantities are calculated by 
adding one-fourth to the quantities in the third column, to give approximate estimate of 
the volume occupied in waggons by the goods, or the space required to load a ton of each 
kind. Sand, No. 39, lies solid in any situation. 



WEIGHT AND SPECIFIC GRAVITY OF LIQUIDS. 



215 



TABLE No. 68.-WEIGHT AND SPECIFIC GRAVITY OF 

LIQUIDS. 



LIQUIDS AT 33* F. 



Mercury 

Bromine. 

Sulphuric acid, maximum concentration.. 

Nitrous acid 

Chloroform. 

Water of the Dead Sea 

Nitric acid, of commerce 

Acetic acid, maximum concentration 

Milk. 

Sea water, ordinary 

Pure water (distilled) at 39°.! F 

Wine of Bordeaux. 

Do. Burgundy 

Oil, lintseed 

j Do. poppy 

1 Do. rape-seed 

Do. whale 

Do. olive 

Do. turpentine '. 

Do. potato 

Petroleum 

Naphtha 

Ether, nitric 

Do. sulphurous 

Do. nitrous 

Do. acetic 

Do. hydrochloric 

Do. sulphuric 

Alcohol, proof spirit 

Do. pure 

Benzine 

Wood spirit 



Weight of one 
cubic foot. 



pounds. 
848.7 
185.I 
1 14.9 
96.8 

95.5 

774 
76.2 

67.4 

64.3 
64.05 

62.425 

62.1 

61.9 

58.7 
58.1 

57.4 

57.4 

57.1 

54.3 
51.2 

54.9 

69.3 
67.4 
55.6 
55.6 

54.3 
44.9 
57.4 
49-3 
53.1 
49-9 



Weight of 
one gallon. 



pounds. 
136.0 . 
29.7 
18.4 , 

15.5 
15.3 . 

12.4 

12.2 . 
10.8 

10.3 , 
10.3 

1 0.0 . 

9-9 
9-9 • 
9-4 

9-3 . 
9.2 

9.2 . 

8.7 . 
8.2 

8.8 . 

8.5 
II. I , 

10.8 

8.9 . 
8.9 
8.7 , 
7.2 

, 9.2 

7.9 
, 8.5 

8.0 



Specific 
Gravity. 



Water = i. 
..13.596 
2.966 
.. 1.84 

1.55 

.. 1.53 
1.24 

.. 1.22 

1.08 
.. 1.03 

1.026 
.. I.OOO 

0.994 
.. 0.991 

0.94 

.. 0-93 
0.92 

.. 0.92 

0.915 
.. 0.87 

0.82 
.. 0.88 

0.85 
.. I. II 

1.08 
.. 0.89 

0.89 
.. 0.87 

0.72 
.. 0.92 

0.79 
.. 0.85 

0.80 



2l6 



WEIGHT, ETC., OF GASES AND VAPOURS. 



TABLE No. 69.— WEIGHT AND SPECIFIC GRAVITY OF 

GASES AND VAPOURS. 



GASES AT 32* F. AND UKDBR ONE 
ATMOSPHERE OP PRESSURE. 



Vapour of mercury (ideal) 

Vapour of bromine 

Chloroform 

Vapour of turpentine 

Acetic ether 

Vapour of benzine 

Vapour of sulphuric ether .... 

Vapour of ether (.'*) 

Chlorine 

Sulphurous acid 

Alcohol 

Carbonic acid (actual) 

Do. (ideal) 

Oxygen 

Air 

Nitrogen 

Carbonic oxide 

Olefiant gas 

Gaseous steam 

Ammoniacal gas 

Light carburetted hydrogen .. 

Coal-gas (page 458) 

Hydrogen 



Volume of one 
pound weight. 



cubic feet. 

.. 1.776... 

2.236 
.. 2.337... 

2.637 

.. 4*075 ••• 

4.598 
.. 4.790 ... 

4.777 
.. 5.077 ... 

5.513 
. . 7.679 . . . 

8.IOI 
.. 8.160 ... 

r 1.205 
.. 12.307 ••• 

12.727 
.. 12.004 ••• 

12.580 

• .19.913 ... 
21.017 

..22.412 ... 

28.279 

179.00 ... 



Weight of one cubic foot. 



in pounds. 
...0.563 .. 

0.447 
...0.428 ... 

0.378 
...0.245 " 

0.217 
...0.209 ... 

0.206 
...0.197 ... 

O.1814 

...ai302 ... 

0.12344 
...0.12259 

0.089253 
...0.080728 

0.078596 
...0.0781 ... 

0.07808 

.0.05022 

0.04758 
...0.04462 

0.03536 
...0.005594 



m ounces. 
..9.008 . 

7.156 
.6.846 . 
6.042 

.3-927 . 
3.480 

.3.340 . 
3.302 

.3.152 . 
2.902 
...2.083 • 

1.975 
.. .1.961 . 

1.428 
.. .1.29165 

1.258 
...1.250 . 

1.249 
...0.8035 • 

0.7613 
...0.7139.. 

0.5658 
...0.0895 



Specific 
Gravity. 



Air = X. 
.. 6.9740 

5.5400 
.. 5.3000 

4.6978 
.. 3.0400 

2.6943 
.. 2.5860 

2.5563 
.. 2.4400 

2.2470 
.. I.6130 

1.5290 
.. 1. 5180 

1. 1056 
.. I.OOOO 

0.9736 
.,0.9674. 

0.9672 
..0.6220 

0.5894. 

.0.5527 
0.4381 

..0.0692 



TABLES OF THE WEIGHT OF IRON AND 

OTHER METALS. 



Wrought Iron. — According to Table No. 65 of the Weight and Specific 
Gravity of Solids, the weight of a cubic foot of wrought iron varies, for 
various qualities, from 466 pounds to 487 pounds per cubic foot, and the 
average weight, taken for purposes of general calculation, is 480 pounds per 
cubic foot. This average weight is equivalent to a weight of 40 pounds per 
square foot, i inch in thickness — a convenient unit, which is usually 
employed in the development of tables of weights of iron for engineering 
and manufacturing purposes. The extremes of variation from this medium 
unit, extend from ^^ pound less, to about fi pound more than 40 pounds 
per square foot, or from 2.2 to 1.5 per cent, either way — a deviation, the 
extent of which is of little or no practical consequence, and which, at all 
events, is comprehended in the percentages allowed in the framing of 
estimates. 

The average weight of a cubic inch of ^vrought iron is 

i?^ =.277 pound, 
1720 

or one-tenth more than a quarter of a pound. For a round number, when 
cubic inches are dealt with, it may be, and is usually, taken as .28 pound, 
which is only four-fifths of i per cent, more than the medium weight, and 
corresponds to a weight of 483.84 pounds per cubic foot, or to 40.32 
pounds per square foot, i inch thick, or to 10 pounds per lineal yard, 
I inch square. 

The volume of i pound of wTought iron is 3.6 cubic inches. 

Sied. — ^The weight of a cubic foot of steel varies from 435 pounds to 
493 pounds per cubic foot, and the average weight is about 490 pounds 
per cubic foot. For convenience of calculation, the average weight is taken 
in the following tables, as 489.6 pounds per cubic foot, for which the 
specific weight is 1.02, when that of wrought iron = 1.00. The weight of a 
square foot, i inch thick, is 40.8 pounds; of a lineal yard, 10.2 pounds; 
and of a cubic inch, .283 pound. 

The volume of i pound of steel is 3.53 cubic inches. 

Cast Iron, — The weight of a cubic foot of cast iron varies from 378^^ 
pounds to 467^ pounds per cubic foot, and the average weight is taken as 
450 ix)unds. The weight of a square foot, i inch thick is, therefore, 37.5 
pounds; of a lineal yard, i inch square, 9.375 pounds; and a cubic inch, 
.26 pound. The specific weight is .9375. 

The volume of i pound of cast iron is 3.84 cubic inches. 

The following data, for the weight of iron, are abstracted for readiness 
of reference: — 



2l8 WEIGHT OF METALS. 

Wrought Iron, Rolled. 

I cubic foot, 480 pounds, or 4.29 cwts. 

I square foot, i inch thick, 40 pounds. 

I square foot, 3 inches thick, 120 pounds, or 1.07 cwts. 

. 3 square feet, i inch thick, 120 pounds, or 1.07 cwts. 

I lineal foot, i inch square, 3 ^ pounds, or .03 cwt 

I cubic inch, say 0.28 pound. 

3.6 cubic inch, i pound. 

I lineal yard, i inch square, 10 pounds. 

I lineal foot, 3 inches square, 30 pounds. 

I lineal foot, 6 inches square, 120 pounds, or 1.07 cwts. 

I lineal foot, 3 inches by i inch thick, 10 pounds. 

I lineal foot, ]/i inch in diameter,.... 2 pounds. 

1 lineal foot, 2 inches in diameter,... 10.5 pounds. 
I lineal foot, 6 J^ in. in diameter, about i cwt 

Cast Iron. 

I cubic foot, 450 pounds, or 4 cwts. 

5 cubic feet, i ton. 

I square foot, i inch thick, 37.5 pounds. 

I squarefoot, 3 inches thick (^ cub. ft), 112.5 pounds, or i cwt 

3 square feet, i inch thick, 112.5 pounds, or i cwt 

I cubic inch, 0.26 pound. 

3.84 cubic inches, i pound. 

The Table No. 70 contains the weight of iron and other metals for the 
following volumes : — 

I cubic foot 

I square foot, i inch thick, or ^ji^ih. of a cubic foot 
I lineal foot, i inch square, or 7xath of a square foot 
I cubic inch, or Viath of a lineal foot. 

A sphere, i foot in diameter. 

The specific gravity due to the respective weights per cubic foot is also 
given, and likewise the specific weight or heaviness, taking the weight of 
wrought iron as i, or unity. 

The next Table, No. 71, contains the volumes of iron and other metals 
for the following weights : — 

I ton, in cubic feet 

I cwt, in square feet, i inch thick. 

I cwt, in lineal feet, i inch square. 

I pound, in cubic inches. 

I ton, as a sphere, in feet of diameter. 

I ton, as a cube, in feet of lineal dimension. 

The next Table, No. 72, contains the weight of 1 square foot of metals of 
various thickness, advancing by sixteenths and by twentieths of an inch, up 
to I inch in thickness. 

The fourth Table, No. 73, contains the weight of prisms or bars of iron, 
and other metals, or metals of any other uniform section, for given sectional 
areas, varying from .1 square inch to 10 square inches of section, advancing 
by one-tenth of an inch, for i foot and i yard in length. 



TABLES Ot WEIGHT AND VOLUME OF METALS. 



219 



This table is useful in calculations of the weights of bars of every form, 
rails, joists, beams, girders, tubes, or pipes, &c., when the sectional area 
is given. 

The table is available for finding the weight of a metal for any sectional 
area up to 100 square inches, by simply advancing the decimal points one 
place to the right; or, in round numbers, up to 1000 square inches, by 
advancing the decimal points two places. For example, to find the weight 
of wrought iron having a sectional area of 17 square inches:— 

For 1.7 square inches, the weight per foot is 5.67 pounds. 
For 17 square inches, the weight per foot is 56.7 pounds. 
For 170 square inches, the weight per foot is 567 pounds. 



Table No. 70. — Weight of Metals. 



Mbtau 



Wrought Iron. 

Cast Iron 

1 Steel 

j Copper, Sheet 

Copper, Hammered 

Tin 

Zinc 

Lead. 

Brass, Cast. 

Brass, Wire, 

Gon Metal 

Silver 

Gold 

Platinum 



Cubic Foot. 



lbs. or cwts. 



480 or 

450 or 
'489.6 or 

549 or 

556 or 

462 or 

437 or 

712 or 

505 or 

533 or 

524 or 

655 or 

1 200 or 

1342 or 



4.29 
4.02 

4-37 
4.90 

4.96 
4.13 
3.90 
6.36 

4.51 
4.76 

4.68 

5.85 
10.72 

12.00 



Square Foot, 
X inch Thick. 



lbs. or cwts. 



40 or 
37.5 or 
40. 8 or 
45.8 or 

46.3 or 

38.5 or 

36.4 or 

59. 3 or 
42. 1 or 

44.4 or 

43- 7 or 

54. 6 or 
loo.oor 
iii.8or 



357 
335 
364 
409 

413 
344 
325 
530 
375 
396 

390 
488 

893 
1. 000 



Lineal 

Foot, 

I Inch 

Square. 



lbs. 

3.333 
3.125 

3.400 

3-813 
3.861 

3.208 

3.035 
4.944 
3.507 
3.701 

3.639 

4.549 

8.333 
9.320 



Cubic 
Inch. 



lb. 

.278 
.260 
.283 
.318 
.322 
.268 

•253 
.412 

.292 

.308 

■304 

.379 
.694 

.777 



Sphere, 
I Foot 
Dia- 
meter.' 



lbs. 

251 
236 
257 
287 
291 
242 
229 

373 
264 

279 

274 

343 
628 

703 



Specific 
Gravity. 



Water 



= x. 



7.698 
7.217 

8.805 
8.917 

7.409 
7.008 

II. 418 
8.099 
8.548 
8.404 

10. 505 

19.245 
21.522 



Specific 
eight. 



apei 
Wei 



Wro'ght 
Iron=z. 

1.000 

.9375 
1.020 

1. 144 

1. 158 

.962 

.910 

1.483 
1.052 

I.IIO 

1.092 

1.365 

2.500 
2.796 



Table No. 71. — ^Volume of Metals for given Weights. 



Metal. 



Wromgfht Iron 

Cast &on 

Steel 

Copper, Sheet .... 
Copper, Hammered 

Tm 

Zinc 

Lead 

Brass, Cast.. 

Brass, Wire. 

Gun Metal 

Silver 

Gold 

Platinum 



Cubic Feet 
to a Ton. 



cubic feet 
4.67 

4.98 
4.58 
4.08 

4.03 
4.86 

5.13 

3.15 

4.44 
4.20 

4.28 

3.42 
1.87 
1.67 



Square Feet, 

X Inch Thick, 

toacwt. 



square feet. 
2.80 
2.99 

2.75 

2.44 
2.42 

2.91 

3.08 

1.89 

2.67 

2.30 

2.56 

2.05 

1. 12 

I.OO 



Lineal Feet, 

I In. Square, 

to a cwt. 



feet. 

33.6 

35.8 

32.9 
29.4 

29.0 

34.9 
36.8 

22.7 

31.9 
30.3 
30.8 
24.6 

13.4 
12.0 



Cubiclnches 
to a lb. 



cubic inches. 
3.60 
3.84 
3.53 
3.15 
3." 
3-74 

3-95 

2.43 
3.42 

3.24 
3-30 

2.64 

1.44 

1.29 



Diameter 

of a Sphere 

of I Ton. 



feet. 
2.07 
2.12 
2.26 
1.98 
1.98 
2.10 
2.14 
1. 81 
2.04 
2.00 
2.02 
1.87 

1.59 
1.47 



Side of a 

Cube of 

z Ton. 



feet. 
1.67 
I.71 

1.66 
1.60 

1-59 
1.69 

1.73 

1.47 
1.64 

1. 61 

1.62 

1.51 

1.28 

1.19 



220 



WEIGHT OF METALS. 



Table No. 72. — Weight of i Square Foot of Metals. 

Thickness advancing by Sixteenths of an Inch. 



Thick- 
ness. 



inch. 

3/16 
% 

s/16 

H 

7/16 

9/16 
H 

"As 
H 

'3/16 

H 

15/16 

I 



Wro't 
Ikon. 

Specific 
wt. =1, 



lbs. 
2.50 
5.00 
7.50 
10. 

12.5 

150 

17.5 
20.0 

22.5 
25.0 
27.5 
30.0 

32.5 
35.0 

37.5 
40.0 



Cast 
Iron. 

Specific 
wt.=.937S. 



lbs. 

2.34 
4.69 

7.0^ 

9.38 

II.7 
14. 1 
16.4 
18.7 

21. 1 

23.5 
25.8 

28.1 

30.5 
32.8 

35-2 
37.5 



Stbel. 
Specific 

Wt.= 1.02. 



lbs. 

2.55 
5.10 
7.65 
10.2 

12.8 

»5-3 
17.9 

20.4 
23.0 

28.1 
30.6 

33.2 

35-7 

38.3 
40.8 



Copper. 

Specific 
wt.=i,i6. 



lbs. 
2.89 

5-79 
8.68 

II. 6 

14.5 

17.4 
20.3 

23.2 

26.0 
28.9 
31.8 
34.7 

37.6 
40.5 

43.4 
46.3 



Tin. 

Specific 
wt=.962. 



lbs. 
2.41 
4.81 
7.22 
963 

12.0 

14.4 
16.8 

19.3 

21.7 
24.1 
26.5 
28.9 

313 

33-7 
36.1 
38.5 



Zinc. 

Specific 
wt.=.9io. 



lbs. 
2.28 

4.55 
6.83 

9.10 

II.4 

137 

15.9 
18.2 

20.5 
22.8 
25.0 

27.3 

29.6 

31-9 

34.1 
36.4 



Brass. 

Specific 
wt. =1.052. 



lbs. 

2.63 

5.26 

7.89 

10.5 

13.2 

il8 
18.4 
21. 1 

237 
26.3 

28.9 

31.6 

34.2 
36.8 

39-5 
42.1 



Gun 
Metal. 

Specific 
wt.= 1.092. 



lbs. 

2.73 
5.46 
8.19 
10.9 

13-7 
16.4 

19. 1 

21.9 

24.6 

27.3 
30.0 

32.8 

35.0 
38.2 
41.0 

43-7 



Lead. 

Specific 
wt.=i.48. 



lbs. 

3.71 
7.41 
II. I 

14.8 

18.5 
22.2 
25.9 
29.7 

33-4 

37. i 
40.8 

44.5 

48.2 

519 
55-6 

59-3 







Thickness advancing by Twentieths of 


an Inch. 






inch. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


.05 


2.00 


1.88 


2.04 


2.32 


'•§3 


1.82 


2. II 


2.19 


2.96 


.10 


4.00 


3.75 


4.08 


4.63 


3.85 


3-64 


4.21 


4.37 


5.93 


.15 


6.00 


5.63 


6.12 


6.95 


5.78 


5.46 


6.32 


6.56 


8.90 


.20 


8.00 


7.50 


3.16 


9.26 


7.70 


7.28 


8.42 


8.74 


"•§ 


.25 


10.0 


9.38 


10.2 


II. 6 


9.63 


9.10 


10.5 


10.9 


14^ 


.30 


12.0 


"3 


12.2 


13.9 


II. 6 


10.9 


12.6 


131 


17.8 


.35 


14.0 


131 


14.3 


16.2 


135 


12.7 


14-7 


15.3 


20.8 


.40 


16.0 


15.0 


16.3 


18.5 


154 


14.6 


16.8 


17.5 


23.7 


.45 


18.0 


16.9 


18.4 


20.8 


17.3 


16.4 


18.9 


19.7 


26.7 


.50 


20.0 


18.8 


20.4 


23.2 


19-3 


18.2 


21. 1 


21.9 


29.7 


.55 


22.0 


20.6 


22.4 


25-S 


21.2 


20.0 


23.2 


24.0 


32-7 


.60 


24.0 


22.5 


245 


27.8 


23.1 


21.8 


25.3 


26.2 


35.6 


.65 


26.0 


24.4 


26.5 


30.1 


25.0 


23-7 


27.4 


28.4 


38.6 


.70 


28.0 


26.3 


28.6 


32.4 


27.0 


255 


29.5 


30.6 


41.5 


•75 


30.0 


28.1 


30.6 


34.7 


28.9 


27.3 


31.6 


32.8 


44-5 


.80 


32.0 


30.0 


32.6 


37.0 


30.8 


29.1 


33-7 


35.0 


47.5 


.85 


34.0 


31-9 


34.7 


39.4 


32.7 


30.9 


35.8 


37.2 


50.4 


.90 


36.0 


33.8 


^tl 


41.7 


34.7 


32.8 


37-9 


39.3 


53-4 


.95 


38.0 


35-6 


38.8 


44.0 


36.6 


34-6 


40.0 


41.5 


56.3 


I.OO 


40.0 


37.5 


40.8 


46.3 


38.5 


36.4 


42.1 


43-7 


59.3 



Note to Table *j% next page. — To find the weight of I lineal foot or I lineal yard of 
hammered iron, copper, tin, zinc, or lead, multiply the tabular weight for rolled wrought 
iron of the given dimensions by the following multipliers, respectively : — 

Exact. Approximate. 

Hammered Iron 1.008 i.oi equivalent to I percent more. 

Copper 1. 158 1. 16 ,, 16 ,, more. 

Tin 962 96 ,, 4 ,, less. 

Zinc 91 91 ,, 9 „ less. 

Lead I'483 1.48 ,, 48 „ more. 



WEIGHT OF METALS OF A GIVEN SECTIONAL AREA. 221 

Table No. 73. — Weight of Metals, of a given Sectional Area, 
PER Lineal Foot and per Lineal Yard. 





Rolled 
Wrought Iron. 


Cast Iron. 


Stesl. 


Brass. 


Gun Mbtal. 


Sect. 
' Akea 


Sp. Wcight=x. 


Sp.Weight=.9375. 


Sp. Weight=i.o2. 


Sp.Weight-=i.o52. 


Sp» Weight= 1.092. 


1 


I Foot. 


I Yard. 


X Foot. 


I Yard. 


I Foot. 


I Yard. 


I Foot 


1 Yard. 


I Foot.* 


I Yard. 


, sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


.1 


-333 


I.OO 


•313 


.938 


.340 


1.02 


•351 


1.05 


■364 


1.09 


.2 


.667 


2.00 


.625 


1.88 


.680 


2.04 


.701 


2.10 


.728 


2.18 


•3 


I.OO 


3.00 


.935 


2.81 


1.02 


3.06 


1.05 


3.16 


1.09 


3.28 


.4 


1-33 


4.00 


1.25 


3.75 


1.36 


4.08 


1.43 


4.21 


1.46 


4-37 


•5 


1.67 


5.00 


1.56 


4.69 


1.70 


5- 10 


1.75 


5.26 


1.82 


5.46 


.6 


2.00 


6.00 


1.88 


5.63 


2.04 


6.12 


2.II 


6.31 


2.18 


6.55 


.7 


2-33 


7.00 


2.19 


6.56 


2.38 


7.14 


2.46 


736 


2.55 


7.64 


.8 


2.67 


8.00 


2.50 


7.50 


2.72 


8.16 


2.81 


8.42 


2.91 


8.74 


•9 


3.00 


9.00 


2.81 


8.44 


3.06 


9.18 


3.16 


9-47 


3.28 


9.83 


1 1.0 


3.33 


10. 


3.15 


9.38 


3.40 


10.2 


3.51 


I0.5 


364 


10.9 


I.I 


3-67 


II.O 


3-44 


10.3 


3-74 


II. 2 


3.86 


II. 6 


4.00 


12.0 


1.2 


4.00 


12.0 


3-75 


"3 


4.08 


12.2 


4.21 


12.6 


4-37 


13.1 


1-3 


4.33 


130 


4.06 


12.2 


4.42 


13-3 


4,56 


13-7 


4.73 


14.2 


1.4 


4.67 


14.0 


4.38 


13- 1. 


4.76 


14.3 


4.91 


14.7 


5.10 


15.3 


i-S 


5.00 


15.0 


4.69 


14. 1 


S'lo 


15-3 


5.26 


15.8 


5.46 


16.4 


1.6 


5.33 
5.67 


16.0 


5.00 


15.0 


5-44 


16.3 


5.61 


16.8 


5.82 


>7-5 


1-7 


17.0 


5.31 


159 


5.78 


17.3 


5.96 


17.9 


6.19 


18.6 


1.8 


6.00 


18.0 


563 


16.9 


6.12 


18.4 


6.31 


18.9 


6.55 


19-7 


1-9 


6.33 


19.0 


5-94 


17.8 


6.46 


19.4 


6.66 


20.0 


6.92 


20.8 


2.0 


6.67 


20.0 


6.25 


18.8 


6.80 


20.4 


7.01 


21.0 


7.28 


21.8 


2.1 


7.00 


21.0 


6.56 


19.7 


7.14 


21.4 


736 


22.1 


7.64 


22.9 


2.2 


7.33 


22.0 


6.88 


20.6 


7.48 


22.4 


7.72 


23.1 


8.01 


24.0 


2-3 


7.67 


23.0 


7.19 


21.6 


7.82 


235 


8.07 


24.2 


8.37 


25.1 


2.4 


8.00 


24.0 


7.50 


22.5 


8.16 


24.5 


8.42 


253 


8.74 


26.2 


2.5 


f.33 


25.0 


7.81 


23-4 


8.50 


255 


8.77 


26.3 


9.10 


27.3 


2.6 


8-67 


26.0 


8.13 


24.4 


8.84 


26.5 


9.12 


27.4 


9.46 


28.4 


H 


9.00 


27.0 


8.44 


253 


9.18 


27.5 


9.47 


28.4 


9.83 


29.5 


2.8 


9.33 


28.0 


8.75 


26.3 


9.52 


28.6 


9.82 


29.5 


10.2 


30.6 


2.9 


9.67 


29.0 


9.06 


27.2 


9.86 


29.6 


10.2 


30s 
31.6 


10.6 


^H 


3.0 


lO.O 


30.0 


9.38 


28.1 


10.2 


30.6 


10.5 


10.9 


32.8 


31 


10.3 


310 


9.69 


29.1 


10.5 


31.6 


10.9 


32.6 


"3 


33-9 


3-2 


10.7 


32.0 


10.0 


30.0 


10.9 


32.6 


II. 2 


33-7 


11.7 


34.9 


3-3 


II.O 


330 


10.3 


309 


II. 2 


33-7 


I1.6 


34.7 


12.0 


36.0 


3-4 


"•3 


34.0 


10.6 


31-9 


11.6 


34.7 


11.9 


35.8 


12.4 


37.1 


3.5 


1 1.7 


35.0 


10.9 


32.8 


11.9 


35-7 


12.3 


36.8 


12.7 


38.2 


3.6 


12.0 


36.0 


"3 


33.8 


12.2 


36.7 


12.6 


37.9 


13.1 


39.3 


3.7 


*2.3 


37.0 


11.6 


34.7 


12.6 


37.7 


130 


38.9 


13.5 


40.4 


3.8 


12,7 


380 


11.9 


35.6 


12.9 


38.8 


133 


40.0 


13.8 


41.5 


3.9 


13.0 


39.0 


12.2 


36.6 


133 


39-8 


13.7 


41.0 


14.2 


42.6 


4.0 


»3.3 


40.0 


12.5 


37.S 


13.6 


40.8 


14.0 


42.1 


14.6 


43.7 


4.1 


13.7 


41.0 


12.8 


38.4 


139 


41.8 


14.4 


43-1 


14.9 


44.8 


4-2 


14.0 


42.0 


131 


39-4 


14.3 


42.8 


14.7 


44.2 


15.3 


45.9 


4.3 


H3 


43.0 


^H 


40.3 


14.6 


43.9 


15. 1 


45.2 


15.7 


46.9 


4-4 


14.7 


44.0 


13-8 


41.3 


15.0 


44.9 


^H 


46.3 


16.0 


48.0 


4-5 


15.0 


45.0 


14. 1 


42.2 


15.3 


45-9 


15.8 


4Z-3 


16.4 


49.1 


4.6 


15.3 


46.0 


14.4 


43-1 


15.6 


46.9 


16. 1 


48.4 


16.7 


50.2 


4.7 


15.7 


47.0 


14.7 


44.1 


16.0 


47.9 


16.5 


49.4 


17. 1 


51.3 


4.8 


16.0 


48.0 


15.0 


45.0 


16.3 


49.0 


16.8 


50.5 


17.5 


52.4 


4.9 


16.3 


49.0 


15-3 


45.9 


16.7 


50.0 


17.2 


51.6 


17.8 


53.5 


5.0 


16.7 


50.0 


15.6 


46.9 


17.0 


51.0 


17.5 


52.6 


18.2 


54.6 



222 



WEIGHT OF METALS. 
Table No 73 {continued). 





R0M.RD 

Wrought Irom. 


Cast Iron. 


Stbel. 


Brass. 


Gun Metal. 


Sect. 


Sp. Weight=x. 


Sp.Weight — 9375- 


Sp.Weight=z.03. 


Sp.Weight=x.o53. 


Sp.Weight=x.o9a. 


Akra 






















mmW^M»*\» 


xFoot 


xYard. 


I Foot 


xYard. 


I Foot 


xYard. 


I Foot 


xVard. 


X Foot 


I Yard. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


5.1 


17.0 


51.0 


15.9 


47.8 


17.3 


52.0 


17.9 


53-7 


18.6 


m 


5.2 


17.3 


52.0 


16.3 


48.8 


17.7 


53- 


18.2 


54- Z 


18.9 


5-3 


17.7 


53.0 


16.6 


49.7 


18.0 


54.1 


18.6 


55.8 


X9-3 


57-9 


5.4 


18.0 


54.0 


16.9 


50.6 


18.4 


55.1 


18.9 


56.8 


19.7 


58.9 


55 


18.3 


55.0 


17.2 


51.6 


18.7 


56.1 


19.3 


57.9 
58.9 


20.0 


60.0 


5.6 


18.7 


56.0 


17.5 


.52.5 


19.0 


57-^ 


19.6 


20.4 
20.8 


61. 1 


5.7 


19.0 


57.0 


17.8 


53.4 


19.4 


58.1 


20.0 


60.0 


62.2 


5.8 


19.3 


58.0 


18. 1 


54-4 


19.7 


59.2 


20.3 


61.0 


21.1 


633 


5-9 


19.7 


59.0 


18.4 


55.3 


20.1 


60.2 


20.7 


62.1 


21.5 
21.8 


64.4 


6.0 


20.0 


60.0 


18.8 


56.3 


20.4 


61.2 


21.0 


63.1 


65.5 


6.1 


20.3 


61.0 


19. 1 


57.2 


20.7 


62.2 


21.4 


64.2 


22.2 


66.6 


6.2 


20.7 


62.0 


19.4 


58.1 


21. 1 


63.2 


21.7 


65.2 


22.6 


67.7 


6.3 


21.0 


63.0 


197 


59.1 


21.4 


64.3 


22.1 


66.3 


22.9 


68.8 


6.4 


21.3 


64.0 


20.0 


60.0 


21.8 


65.3 


22.4 


67.3 


23.3 


69.9 


6.5 


21.7 


65.0 


20.3 


60.9 


22.1 


66.3 


22.8 


68.4 


23.7 


70.9 


6.6 


22.0 


66.0 


20.6 


61.9 


22.4 


67.3 


23.1 


69.4 


24.0 


72.0 


H 


22.3 


67.0 


20.9 


62.8 


22.8 


68.3 


23.5 


70.5 


24.4 
24.8 


731 


6.8 


22.7 


68.0 


21.3 


63.8 


23.1 


69.4 


23.9 


71.5 


74.2 


6.9 


23.0 


69.0 


21.6 


64.7 


23.5 


70.4 


24.2 


72.6 


25.1 


75-3 


7.0 


23.3 


70.0 


21.9 


65.6 


23.8 


71.4 


24.6 


73.6 


25.5 


76.4 


7.1 


23.7 


71.0 


22.2 


66.6 


24.1 


72.4 


24.9 


74.7 


25.8 


77-5 


7.2 


24.0 


72.0 


22.5 


67.5 


24,5 


73.4 


25.3 


75-7 


26.2 


78.6 


7.3 


24.3 


730 


22.8 


68.4 


24.8 


74.5 


25.6 


76.8 


26.6 


IH 


7.4 


24.7 


74.0 


23.1 


69.4 


25.2 


75.5 


26.0 


77.9 


26.9 


80.8 


7.5 


25.0 


75.0 


^H 


70.3 


25.5 


76.5 


26.3 


78.9 


27.3 


81.9 


7.6 


253 


76.0 


23.8 


71.3 


25.9 


77-5 


26.7 


80.0 


27.7 


83.0 


7.7 


25.7 


77.0 


24.1 


72.2 


26.2 


78.5 


27.0 


81.0 


28.0 


84.1 


7.8 


26.0 


78.0 


24.4 


73.1 


26.5 


79.6 


27.4 


82.1 


^?i 


85.2 


7.9 


26.3 


79.0 


24.7 


74.1 


26.9 


80.6 


27.7 


83.1 


28.8 


86.3 


8.0 


26.7 


80.0 


25.0 


75.0 


27.2 


81.6 


28.1 


84.2 


29.1 


87.4 


8.1 


27.0 


81.0 


253 


75-9 


27.5 


82.6 


28.4 


85.2 


295 


88.5 


8.2 


27.3 


82.0 


25.6 


76.9 


27.9 


83.6 


28.8 


86.3 


29.9 


89.5 


8.3 


27.7 


83.0 


25.9 


77.8 


28.2 


84.7 


29.1 


111 


302 


90.6 


8.4 


28.0 


84.0 


26.3 


78.8 


28.6 


85.7 


29.5 
29.8 


30-6 


91.7 


!-5 


28.3 


85.0 


26.6 


79.7 


28.9 


86.7 


89.4 


30.9 


92.8 


8.6 


28.7 


86.0 


26.9 


80.6 


29.2 


ll^ 


30.2 


90.S 


31-3 


93-9 


8.8 


29.0 


5Z-° 


27.2 


81.6 


29.6 


IH 


30.5 


91.5 


31.7 


95.0 


29.3 


88.0 


27.5 


82.5 


29.9 


89.8 


30.9 


92.6 


32.0 


96.1 


8.9 


29.7 


89.0 


27.8 


83.4 


30.3 


90.8 


31.2 


93-6 


32.4 


97.2 


9.0 


30.0 


90.0 


28.1 


84.4 


30.6 


91.8 


316 


94.7 


32.8 


98.3 


9.1 


30.3 


91.0 


28.4 


!l-3 


309 


92.8 


31-9 


957 


33.x 


99.4 


9.2 


30.7 


92.0 


28.8 


86.3 


31-3 


93.8 


32.3 


96.8 


33.5 


100.5 


9.3 


31.0 


930 


29.1 


87.2 


31.6 


94.9 


32.6 


97.8 


33.9 


101.6 


9.4 


31.3 


94.0 


29.4 


88. z 


32.0 


95-9 


33.0 


98.9 


34-2 


102.7 


^•5 


31.7 


95.0 


29.7 


89.1 


32.3 


96.9 


33-3 


99.9 


34-6 


X03.7 


9.6 


32.0 


96.0 


30.0 


90.0 


32.6 


97.9 


33.7 


lOI.O 


34.9 


104. S 


H 


32.3 


97.0 


30.3 


90.9 


33'0 


98.9 


34-0 


102.0 


35.3 


105.9 


9.8 


32.7 


98.0 


30.6 


91.9 


33.3 


100. 


34.4 


103. 1 


35.7 


107.0 


9.9 


33.0 


99.0 


30.9 


92.8 


33.7 


lOI.O 


34.7 


104.2 


36.0 


108. 1 


lO.O 


33-3 


100. 


31.3 


93.8 


34-0 


102.0 


35.1 


105.2 


36.4 


109.2 



See note at foot of page 220. 



RULES FOR WEIGHT. 223 

Rules for the Weight of Iron and Steel. 

The following rules for finding the weight of wrought iron, cast iron, 
and steel, are based on the data contained in Tables No. 70 and 71. 

Rule i. — ^To find the Weight of Iron or Steel, when the volume 
in cubic feet is given. Multiply the volume by 

4.29 for wrought iron, 
4.02 for cast iron, 
4.37 for steel. 

The product is the w«ight in hundredweights. 

Rule 2. — When the volume in cubic inches is giveny multiply the volume 
by 

.278 (or .28) for wrought iron, 
.26 for cast iron, 
.283 for steel. 

The product is the weight in pounds. 

Rule 3. — WAen the quantity is reduced to square feet, one inch in thickness^ 
multiply the area by 

40 for wrought iron, 

37/^ for cast iron, 

40.8 (or 41) for steel. 
The product is the weight in pounds. 
Or, multiply the area by 

•357 ^or wrought iron, 
•335 ^or cast iron, 
.364 for steel. 

The product is the weight in hundredweights. 

Rule 4. — When the sectional area in square inches, and t/ie length in feet^ 
of a bar or prism are givcfi, multiply the sectional area by the length, 
and by 

3 V3 for wrought iron, 

3^ for cast iron, 

3.4 for steel. 

The product is the weight in pounds. 

For large masses, multiply the sectional area by the length, and divide 
the product by 

672 for wrought iron, 

717 for cast iron, 

659 for steel. 

The quotient is the weight in tons. 

Rule 5. — W/ien the sectional area in square inches , and the length in yards , 
of a barorprisniy are given, multiply the sectional area by the length, and by 

10 for wrought iron, 
9.375 for cast iron, 
10.2 for steel. 

The product is the weight in pounds. 



224 WEIGHT OF METALS. 

Rule 6. — To find the sectional area of a bar or prism of iron 
OR steel, when the length and tfu toted weight are given. Divide the weight 
in pounds by the length in feet, and by 

3 ^3 for wrought iron, 
3^ for cast iron, 
3.4 for steel. 

The quotient is the sectional area in square inches. 

Rule 7. — To find the length of a bar, prism, or other piece 
of uniform section of iron or steel, whefi the total weight and the 
sectional area are given. Divide the weight in pounds by the sectional area 
in square inches, and by 

3 '/3 for wrought iron, 
3 ^ for cast iron, 
3.4 for steel. 

The quotient is the length in feet. 

In applying the last rule to calculate the length of wire of a given size, 
for a given weight, say i cwt. of wire, the sectional area of the wire is 
found, in the usual way, by multiplying the square of the thickness or 
diameter, //, by .7854. Then, by the rule, the length in feet of i cwt. of 
iron wire is equal to 

112 42.78 

In the same way, the dividends of the fractions to express the length of 
I cwt. of other metals may be found, and the following is a special rule 
lor wire : — 

Rule 8. — To find the length of one hundredweight of wire 
OF A given thickness. Divide the following numbers by the square of 
the diameter or thickness, in parts of an inch : — 

42.78 for wrought iron, 
42 for steel, 
37.43 for copper, 
38.54 for brass, 
31.34 for silver, 
17.12 for gold, 
15.28 for platinum. 

The quotient is the length in feet. 

Note. — This rule may be used for finding the weight of round bar iron. 

2. It is known that the density of wire is not perfectly constant, but 
that there is some degree of variation, according to the size. It is generally- 
understood that the density is reduced as the wire is drawn smaller, but 
it appears from the table of the weight of Warrington wire, that the density 
is greater as the size is less. The same inference is to be drawn from 
tabular statements of the length of one kilogramme of wire according to 
the French gauge (Table No. 31, page 148). One of these statements is 
given on the next page, from which it is apparent that the length of iron 



RULES FOR WEIGHT. 



225 



required to weigh a kilogramme decreases more rapidly than the sectional 
area increases. For example, the diameter being 

6, 12, 24, 30 tenths of a millimetre, 
the squares of which, or the relative volumes of a given length, are as 

I. 4, 16, 25; 
the lengths of a kilogramme are 

405» ii5> 3o> 20 metres, 
which are inversely as 

^ 3-5» i3-5> 20.2. 

Showing that a shorter length is required in proportion to the volume, as 
the diameter of the wire is reduced, and that the density of the smaller 
wire must therefore be the greater. 

Table No. 73^. — ^Weight of Galvanized Iron Wire (French). 



Ka. of Gauge. 


Diameter. 


Length of 
X Kilogramme. | 


No. of Gauge. 


Diameter. 


Length of 
z Kilogramme. 




millimetres. 


metres. 




millimetres. 


metres. 


I 


0.6 


405 


13 


0.20 


40 


2 


0.7 


370 


14 


0.22 


35 


3 


0.8 


260 


15 


0.24 


30 


4 


0.9 


215 


16 


0.27 


25 


5 


O.IO 


175 


17 


0.30 


20 

• 


6 


O.I I 


140 


18 


0.34 


15 


7 


0.12 


115 


19 


0-39 


10 


8 


0.13 


103 


20 


0.44 


9 


9 


0.14 


82 


21 


0.49 


6 


10 


0-15 


70 


22 


0.54 


5 


II 


0.16 


65 


23 


0-59 


4 


12 


0.18 


50 









3. The densities of metals assumed in the foregoing rules are those which 
are tabulated in Table No. 65. 

4. In estimating the weight of cast iron from plans, the weight is fre- 
quently calculated at the same rate as for wrought iron, which is heavier 
^n cast iron, with the object of providing an allowance, by way of com- 
pensation, for occasional swellings or enlargements of castings in excess of 
'^''•e exact dimensions of patterns. 

The following tables of the weight of metals in various forms have been 
'Collated by means of the preceding rules. The sectional areas of bars 
~id other pieces of uniform section are, in some tables, added for each 
^^ntling. The length of bar, and the area of plates and sheets, required 
'"^ weigh I cwt., or i ton, are given. 

10 



226 WEIGHT OF METALS. 

LIST OF TABLES OF THE WEIGHT OF WROUGHT IRON, 

In Bars, Plates, Sheets, Hoop-iron, Wire, and Tubes. 

Table No. 74. — Weight of Flat Bar Iron; width, i to 11 inches; thick- 
ness, 7x6 to I inch; length, i to 9 feet. 

Table No. 75. — Weight of Square Iron; ^ to 6 inches square; length, 
I to 9 feet 

Table No. 76. — ^Weight of Round Iron, ^ to 24 inches in diameter; 
length, I to 9 feet 

Table No. 77.— Weight of Angle-Iron and Tee-Iron; sum of the width 
and depth, ly^ to 20 inches; thickness, J^ to i inch; length, i foot 

In the composition of this table, it has been assumed that the base and 
the web or flange are of equal thicknesses; and that the reduction of area 
of section by rounding off the edges, is compensated by the filling in at 
the root of the flange. 

Table No. 78. — ^Weight of Wrought-iron Plates; area, i to 9 square 
feet; thickness, X to 15 inches. 

Table No. 79. — Weight of Sheet Iron, according to wire-gauge used by 
South Staffordshire sheet-rollers; area, i to 9 square feet; thickness. No. i 
to No. 32 wire-gauge. 

Table No. 80. — Weight of Black and Galvanized Iron Sheets (Morton's 
Table). 

Table No. 81. — Weight of Hoop Iron; width, ^ to 3 inches; thickness. 
No. 4 to No. 21 wire-gauge; length, i foot 

Table No. 82. — ^Weight and Strength of Warrington Iron Wire. 

Table No. 83. — Weight of Wrought-iron Tubes, by internal diameter; 
diameter, f^ to 36 inches; thickness, ^ inch to No. 18 wire-gauge; length, 
I foot 

. Table No. 84. — Weight of Wrought-iron Tubes, by external diameter; 
diameter, i to 10 inches; thickness, No. 15 wire-gauge to s/x6 inch; length, 
I foot 



Multipliers, derived from table No. 70, are subjoined, by which the 
tabulated weights of wrought iron may be multiplied, in order to find from 
these tables the weight of bars, plates, or sheets of other metal. — 

Multipliers. 

Hammered Iron i.oi 

Cast Iron 94 

Steel ; 1.02 

Sheet Copper 1.14 

Hammered Copper 1.16 

Lead 1.48 

Cast Brass 1.05 

Brass Wire i.ii 

Gun Metal 1.09 



FLAT BAR IRON. 



227 



Table No. 74— WEIGHT OF FLAT BAR IRON. 

I INCH Wide. 



Thick- 
xass. 


Sbct. 

AXSA. 








Length in ] 


Feet. 








Len^h 

to weigh 

X cwt. 


z 


2 


3 


4 


5 


6- 


7 


8 


9 


iacbes. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


ll». 


lbs. 


lbs. 


lbs. 


feet 


H 
5/16 

H 

llt6 


-250 
.313 

.438 
.500 


.833 
1.04 
1.25 
1.46 
1.67 


1.67 
2.08 
2.50 
2.92 

3-33 


2.50 
3.12 

3-75 
4.38 
5.00 


3.33 

4.16 
5.00 

5.84 

6.67 


4.17 
5.20 

6.25 

7.29 

8.33 


5.00 
6.24 
7.50 
8.76 

lO.O 


5.83 
7.28 

8.75 
10.2 

II.7 


6.67 
8.32 

10. 

II.7 
13-3 


7.50 
9.36 
"3 

13. 1 
15.0 


134.4 

89.6 
76.8 
67.2 


9/t6 
"/i6 


•5^3 

.750 


1.88 
2.08 
2.29 
2.50 


3.75 
4.16 

4.58 
5.00 


5.62 
6.25 
6.87 
7.50 


7.50 
8.33 
9.17 

lO.O 


9.37 
10.4 

II.4 
12.5 


"•3 
12.5 

13.8 
15.0 


13- 1 
14.6 

16.0 
17.5 


15.0 
16.6 
18.3 

20.0 


16.9 
18.8 
20.6 
22.5 


597 

53-8 
48.9 

44.8 


I 


.813 
.875 

-938 

1. 00 


2.71 
2.92 

3.13 
3.33 


5.42 

5.84 
6.25 

6.67 


8.12 
8.76 

9.38 
10.0 


10.8 

II. 7 
12.5 

13.3 


13.5 
14.6 

15.6 

16.7 


16.3 

18.8 
20.0 


19.0 
20.4 
21.9 

23.3 


21.7 

23.4 

25.0 
26.7 


24.4 
26.3 
28.1 

30.0 


41.4 

38.4 
35.8 
33.6 



i}i INCHES Wide. 



1 

iacbes. 


sq. in. 


V 


.2S1 


S/jfi 


.352 


H 


.422 


7/16 


.492 


H 


.563 


»/t« 


.633 


H 


.703 


r 


.844 


r 


.914 
.984 


'5/16 


1.06 


I 


I.I3 



lbs. 

.938 
I.17 
I.4I 
1.64 
1.88 

2. II 

2.34 
2.58 

2.91 

3.05 
3.28 

3.52 

3-75 



lbs. 

1.88 
2.34 
2.S1 
3.28 

3.75 

422 
4.69 
5.16 

5.63 

6.09 
6.56 

7.03 
7.50 



lbs. 

2.81 

3.52 
4.22 
4.92 
5.62 

6.33 

7.03 

7.73 
8.44 

9.14 

9.84 
10.6 

"•3 



lbs. 

3.75 
4.68 
5.62 
6.56 
7.50 

8.44 
9.38 
10.3 

"3 

12.2 

13. 1 
14. 1 

15-0 



lbs. 

4.69 
5.86 

7.03 
8.20 

9.38 

10.6 

n.7 
12.9 

14.0 

15.2 
16.4 
17.6 
18.8 



lbs. 

5.63 
7.03 
8.44 

9.84 

"3 
12.7 

13.1 

15.5 

16.9 

18.3 
19.7 

21. 1 
22.5 



lbs. 

6.56 
8.20 
9.84 
II.5 

13.1 

14.8 
16.4 
18.0 
19.7 

21.3 
23.0 
24.6 
26.3 



lbs. 

7.50 

9.37 
"•3 
13. 1 
15.0 

16.9 
18.8 
20.6 
22.5 

24.4 
26.3 
28.1 
30.0 



lbs. 

8.44 
10.6 
12.7 
14.8 
16.9 

19.0 
21. 1 
23.2 

25.3 

27.4 
29.5 
31.6 

33.8 



1% INCHES Wide. 




lbs. 

3.12 

3.91 
4.69 

5.47 
6.25 

7.03 
7.81 

8.59 
9.38 

10.2 

10.9 
II. 7 

12.5 



lbs. 

4.17 
5.21 

6.25 

7.29 

8.33 
9.38 

10.4 

II.5 
12.5 

13.5 

14.6 
15.6 
16.7 



lbs. 

5.21 
6.51 
7.81 
9.12 
10.4 

II.7 
13.0 

14.3 
15.6 

16.9 
18.2 

20.8 



lbs. 

6.25 
7.82 

9.38 
10.9 

12.5 

14. 1 

15.6 
17.2 
18.8 

20.3 
21.9 

23.4 
25.0 



lbs. 

7.29 
9. 1 1 
10.9 
12.8 
14.6 

16.4 
18.2 
20.1 
21.9 

237 

25-5 

27.3 
29.2 



lbs. 

8.33 
10.4 

12.5 

14.6 

16.7 

18.8 
20.8 
22.9 
25.0 

27.1 
29.2 
31.2 

33.3 



lbs. 

9.37 
II. 7 

14. 1 

16.4 

18.8 

14. 1 

23.4 

2J.8 

28.1 

30.5 
32.8 

35.1 
37.5 



feel. 

II9.5 
95.6 

79.6 

68.3 

59.7 

53-1 
47.8 

43-4 
39-8 

36.8 

34.1 
31.9 
29.9 



feet. 

107.5 
94.0 

71.7 
61.2 

53.8 

47.8 
43.0 
39.1 

35.8 

33.' 
30.7 
28.7 

26.9 



228 



WEIGHT OF METALS. 



Weight of Flat Bar Iron. 
i)i INCHES Wide. 













Length in Feet. 










Thick- 
ness. 


Sect. 
Area. 




















Len^ 
to weigh 
























z 


a 


3 


4 


5 


6 


7 


8 


9 


X CWL 


inches. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet. 


X 


.344 


1.15 


2.29 


3.44 


4.58 


5.73 


6.87 


8.02 


9.17 


10.3 


97.7 


S/i6 


.430 


1-43 


2.86 


4.30 


5.73 


7.16 


8.59 


10. 


II.5 


12.9 


78.2 


H 


.516 


1.72 


3-44 


5.16 


6.87 


8.59 


10.3 


12.0 


13-7 


'$■5 


65.6 


7/16 


.602 


2.01 


4.01 


6.02 


8.02 


10. 


12.0 


14.0 


16.0 


18.0 


48."9 


}i 


.688 


2.29 


4.58 


6.87 


9.17 


"•5 


13-8 


16.0 


18.3 


2a 6 


9/16 


•273 
.859 


2.58 
2.86 


5.16 


7.73 


10.3 


12.9 


15.5 


18.0 


20.6 

1 


23.2 


43-4 


H 


5.73 


8.59 


II.5 


'4-3 


17.2 


20.1 


22.9 


25.8 


39.1 


r 


.945 


315 


6.31 
6.88 


9-45 


12.6 


15.8 


18.9 


22.1 


25.2 


28.4 


35-5 


1.03 


3.44 


10.3 


13.8 


17.2 


20.6 


24.1 


27.5 


309 


•32.6 


»3/i6 


1. 12 


3-72 


8.02 


II. 2 


14.9 


18.6 


22.3 


26.1 


29.8 


33.5 


30.1 


'A 


1.20 


4.01 


12.0 


16.0 


20.0 


24.1 


28.1 


32.1 


36.1 


27.9 


»s/i6 


1.29 


4.30 


8.59 


12.9 


17.2 


21.5 


25.8 


30.1 


34-4 


38.7 


26.1 


I 


i.3» 


4.58 


9.17 


13.8 


18.3 


22.9 


27.5 


32.1 


36.7 


41.3 


24.4 



i}i INCHES Wide. 



inches. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet. 


X 


.275 


1.25 


•2.50 


3-75 


5.00 


6.25 


7.50 


8.75 


10. 


"•3 


89.6 


s/16 


.469 


1.56 


3.13 


4.69 


6.25 


7.82 


9.38 


10.9 


12-5 


14. 1 


78.3 


H 


.563 


1.88 


3-75 


5.63 


7.50 
8.75 


9.38 


IL3 


13. 1 


150 


16.9 


59.7 


7/16 


.656 


2.19 


4.38 


6.56 


10.9 


13. 1 


15.3 


17.5 


19.7 


5*-; 


H 


.750 


2.50 


5.00 


7.50 


10.0 


12.5 


15.0 


17.5 


20.0 


22.5 


44.8 


9/16 


.844 


2.81 


5.63 


8.44 


"•3 


14. 1 


16.0 


19.7 


22.5 


25.3 


39.8 


^, 


.938 


3.13 


6.25 
6.88 


9.38 


12. J 


15.6 


18.8 


21.9 


25.0 


28.1 


35.8 


r 


1.03 


3-44 


10.3 


13.8 


17.2 


20.6 


24.1 


27.5 


30.9 


32.6 


i»3 


3.75 


7.50 


"3 


15.0 


18.8 


22.5 


26.3 


30-0 


33.8 


29.9 


«3/i6 


1.22 


4.06 


8.13 


12.2 


16.3 


20.3 


24.4 


28.4 


32.5 


36.6 


27.6 


^. 


I-3I 


4.38 


8.75 


131 


17.J 


21.9 


26.3 


30.6 


350 


39.4 


25.6 


»s/i6 


1.41 


4.69 


9-38 


14. 1 


18.8 


23.4 


28.1 


32.8 


37.5 


42.2 


23.9 


I 


1.50 


5.00 


lO.O 


15.0 


20.0 


25.0 


30.0 


350 


40.0 


45.0 


22.4 



i^ INCHES Wide. 



inches. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet. 


X 


.406 


1-35 


2.71 


4.06 


5-41 


6.8 


8.10 


9.48 


10.8' 


12.2 


82.7 


s/16 


.508 


1.69 


3-39 


5.07 


6.77 


8.5 


10.2 


II.8 


13.5 


15.2 


66.2 


H 


.boq 


2.03 


4.06 


6.09 


8.12 


10.2 


12.2 


14.2 


16.2 


18-3 


55.1 


7/16 


.813 


2.37 


4-74 


7.II 
8.12 


9.48 


11.8 


14.2 


16.6 


19.0 


21.3 


47.3 


^ 


2.71 


5.42 


10.8 


13-5 


16.2 


19.0 


21.6 


24-4 


41.3 


9/16 


.914 


3.05 


6.09 


9.14 


12.2 


^i-^ 


18.3 


21.3 


24.4 


27.4 


36.8 


^, 


1.02 


3-39 


6.77 


10.2 


13.5 


16.9 


20.3 


23.7 


^H 


30.5 


33.1 


"A6 


1. 12 


3-72 


7.45 


II. 2 


14.9 


18.6 


22.3 


26.1 


29.8 


33-5 


30.1 


H 


1.22 


4.06 


8.13 


12.2 


16.3 


20.3 


24.4 


28.4 


32.5 


36.6 


27.6 


»3/i6 


1.32 


4.40 


8.80 


132 


17.6 


22.0 


26.4 


30.8 


35-2 


39.6 


25.4 


^. 


L43 


4.74 


9.48 


14.2 


19.0 


23.7 


28.4 


33.2 


37.9 


42.7 


23.6 


^s/t6 


I.S3 


5.08 


10.2 


15.2 


20.3 


254 


30.5 


35.5 


40.6 


48.8 


22.1 


I 


1.63 


5.42 


10.8 


16.3 


21.7 


27.1 


32.5 


37-9 


43.3 


21.2 



FLAT BAR IRON. 



229 



Weight of Flat Bar Iron. 
1^ INCHES Wide. 













Length in 


Fbbt. 










Thioc- 

NESS. 


Sbct. 
Aksa. 




















Length 
to weigh 
























Z 


2 


3 


4 


5 


6 


7 


8 


9 


X cwt. 


inches. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet 


X 


.638 


1.46 


2.92 


4.37 


5.83 


7.29 


8.74 


10.2 


II.7 


13. 1 


76.8 


5/16 


.547 


1.82 


3.6J 


§-^? 


7.29 


9.II 


10.9 


12.8 


14.6 


16.4 


61.4 


H 


.656 


2.19 


4.38 


6.56 


8.75 


10.9 


13.1 


15-3 


ns 


19.7 


51.2 


7/16 


.766 


2.55 


H^ 


7.66 


10.2 


12.8 


15.3 


17.9 


20.4 


23.0 


43.9 


a 


.875 


2.92 


5.«3 


8.75 


II.7 


14.6 


17.5 


20.4 


23.3 


26.2 


38.4 


9/16 


.984 


3.28 


6.56 


9.84 


13. 1 


16.4 


19.7 


23.0 


26.2 


29.5 


34.1 


H 


1.09 


3.65 


7.29 


10.9 


14.6 


19.2 


21.9 


2$-5 


29.2 


32.8 


•30-7 


r 


1.20 


4.01 


8.02 


12.0 


16.0 


20.0 


24.1 


28.1 


32.1 


36.1 


27.9 


I.3I 


4.3« 


8.75 


13.1 


17.5 


21.9 


26.3 


306 


350 


39.4 


25.6 


r 


1.42 
1.53 


4.74 
5.10 


9.48 
10.2 


14.2 
15.3 


19.0 
20.4 


23.7 

25. S 


28.4 
30.6 


33.2 
35-7 


37.9 
40.8 


43-2 
45.9 


23.7 
21.9 


^s/i6 


1.64 


5-47 


10.9 


16.4 


21.9 


27.3 


32.8 


1? ) 1 
40.8 


43-7 


49.2 


20.5 


I 


1.75 


5.«3 


II. 7 


17.5 


233 


29.2 


35.0 


46.7 


52.5 


19.2 



iji INCHES Wide. 



tnf|<^^ 


1 

sq. in. 
' .469 


lbs. 

1.56 


lbs. 
3.13 


lbs. 

4.69 


lbs. 
6.2$ 


lbs. 
7.81 


lbs. 
9.38 


lbs. 
10.9 


lbs. 

12.5 


lbs. 

14. 1 


feet 
71.7 


'4 


5/.6 


1 .586 


1.95 


3-91 


5.86 


7.81 


9.66 


11.7 


13.7 


15.6 


17.6 


57-3 


> H 


.703 


2.34 


4.69 


7.03 


9.37 


II.7 


14. 1 


16.4 


18.8 


21. 1 


47.8 


r/i6 


1 .820 


2.73 


5.47 


8.20 


10.9 


13.7 


16.4 


19.1 


21.9 


24.6 


41.0 


}i 


1 .938 


313 


6.25 


9.38 


12.5 


15-6 


18.8 


21.9 


25.0 


28.1 


35.8 


9/16 


1.06 


3.52 


7.03 


10.5 


14. 1 


17.6 


21. 1 


24.6 


28.1 


31.6 


31.8 


H 


1. 17 


3-9" 


7.81 


II.7 


14.6 


19-5 


23.4 


27.3 


31.2 


35.2 


28.7 


"/•« 


1.29 


430 


8.59 


12.9 


17.2 


21.5 


2J.8 

28.1 


30.1 


34.4 


38.7 


26.1 


' )i 


I.4I 


4.69 


9.38 


14. 1 


18.8 


234 


32.8 


37.5 


42.2 


239 


: »Vi6 


! 1.52 


5.08 


10.2 


15.2 


20.3 


25.4 


30.5 


35-5 


40.6 


45.7 


22.1 


 H 


1 1.64 


t^ 


10.9 


16.4 


21.9 


27.3 


32.8 


38.3 


43.9 


49.4 


20.5 


1 '5/i« 


1.76 


11.7 


17.6 


23.4 


29.3 


35.1 


41.0 


46.9 


52.7 


19. 1 


1' 


J.88 


6.25 


12.5 


18.8 


25.0 


3>.3 


37.5 


43.8 


50.0 


56.2 


17.9 



2 INCHES Wide. 



indies. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet 


j( 


.500 


1.67 


3-33 


5.00 


6.67 


8.33 


10.0 


II.7 


13.3 


15-O 


67.2 


s/i6 


.625 


2.08 


4.17 


6.25 


8.33 


10.4 


12.5 


14.6 


16.7 


18.8 


53.8 


H 


1 .750 


2.50 


5.00 


7.50 


lO.O 


12.5 


15.0 


17.5 


20.0 


22.5 


44.8 


7/.6 


-875 


2.92 


5.83 


8.75 


II.7 


14.6 


17.5 


20.4 


23.3 


26.3 


38.4 


>^ 


I.OO 


3-33 


6.67 


10.0 


133 


16.7 


20.0 


23.3 


26.7 


300 


33-6 


9/i« 


1 1. 13 


3.75 


7.50 


"3 


15.0 


18.8 


22.5 


26.3 


300 


33-8 


29.9 


X, 


1 1.38 


4.17 


8.33 


12.5 
13.8 


16.7 


20.8 


25.0 


29.2 


33.3 


37.5 


26.9 


r 


4.58 


9.16 


18.3 


22.9 


27.5 


32.1 


36.7 


41.2 


24.4 


1.50 


5.00 


10.0 


15.0 


20.0 


25.0 


30.0 


35.0 


40.0 


45.0 


22.4 


«3/«6 


' 1.63 


5J^ 


10.8 


16.3 


21.7 


27.2 


32. s 


37.9 


43.3 


48.8 


20.7 


H 


m 


S.83 


11.7 


'7-5 

18.8 


23.3 


29.2 


35.0 


40.8 


46.7 


52.5 


19.2 


«5/x6 


6.25 


12.5 


25.0 


31.3 


37.5 


43.8 


50.0 


56.3 


17.9 


I 


2.00 


6.67 


13.3 


20.0 


26.7 


33.3 


40.0 


46.7 


52.2 


60.0 


16.8 



230 



WEIGHT OF METALS. 



Weight of Flat Bar Iron. 
2>i INCHES Wide. 













Length in 


Febt. 










Thick- 
ness. 


Sect. 
Area. 




















Len^h 
to weigh 
























z 


2 


3 


4 


5 


6 


7 


8 


9 


I cwt. 


inches. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet 


V 


.531 


1.77 


3.54 


I'V 


7.08 


8.85 


10.6 


12.4 


14.2 


15.9 


63.2 


5/i« 


.664 


2.21 


4.43 


6.64 


8.85 


II.7 


13.3 


^5-5 


17.7 


19.9 


50.6 


H 


.797 


2.66 


5-31 


7.97 


10. D 


13.3 


'5-9 


i8.6 


21.2 


23.9 


42.2 


7/16 


.930 


3.10 


6.20 


930 


12.4 


15.5 


18.6 


21.7 


24.8 


27.9 


36.1 


}i 


1.06 


3.54 


7.08 


10.6 


14.2 


17.7 


21.3 


24.8 


28.3 


31.9 


31.6 


9/16 


1.20 


3.98 


7.97 


12.0 


15.9 


20.0 


23.9 


27.9 


31.9 


35-5 


28.1 


^. 


1.33 


4.43 


8.85 


13.3 


17.7 


22.1 


1 26.6 


31.0 


35.4 


39.8 


25.3 


r 


1.46 


4.87 


9.74 


14.6 


19.5 


24.4 


1 29.2 


34.1 


39- 


43.8 


23.0 


1.59 


5.3« 


10.6 


"59 


21.2 


26.6 


31.9 


37.2 


42.5 


47.8 


21. 1 


»3/x6 


1.74 


5.76 


11.5 


'H 


23.0 


28.8 


34.5 


40.3 


46.0 


51.8 


19.8 


H 


1.86 


6.20 


12.4 


18.6 


24.8 


31.0 


37-; 


43.4 


49.6 


55.8 


18. 1 


»5/i6 


1.98 


6.64 


13.3 


19.9 


26.6 


33.2 


39.8 


46.5 


53.1 


59.8 


16.9 


I 


2.13 


7.08 


14.2 


21.3 


28.3 


35.4 


42.5 


49.6 


56.7 


63.8 


15.8 



2)4 INCHES Wide. 



2f^ inches Wide. 



inches. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet. 


X 


.563 


1.88 


3.75 


5.63 


7.50 


9.4 


"3 


13-1 


15.0 


16.9 


59.7 


5/16 


.703 


2.34 


4.69 


7.03 


9.38 


II.7 


14. 1 


16.4 


18.8 


21.1 


47.8 


^ 


.844 


2.81 


5.63 


8.44 


"3 


14. 1 


16.9 


19.7 


22.5 


253 


39.8 


7/16 


.984 


3.28 


6.56 


9.84 


13. 1 


16.4 


19.7 


23.0 


26.3 


29.5 
33.8 


34.1 


^ 


I.I3 


3.75 


7.50 


11.3 


15.0 


18.8 


22.5 


26.3 


30.0 


29.9 


9/16 


1.27 


4.22 


8.44 


12.7 


16.9 


21. 1 


^5-3 


29.5 


33.8 


38.0 


26.5 


^, 


I.4I 


4.69 


9.38 


14. 1 


18.8 


23.4 


28.1 


32.8 


37.5 


42.2 


23.9 


r 


1.55 


5.16 


10.3 


15.5 


20.6 


25.8 


30.9 


36.1 


41.3 


46.4 


21.7 


1.69 


5.63 


"3 


16.9 


22.5 


28.1 


33.8 


39.4 


4S.O 


50.6 


19,9 


»3/i6 


1.83 


6.09 


12.2 


18.3 


24.4 


30.5 


36.6 


42.7 


48.8 


54.9 


18.4 


^. 


1.97 


6.56 


13. 1 


19.7 


26.3 


32.8 


39.4 


45.9 


52.5 


59.1 


17.I 


»5/i6 


2.II 


7.03 


14.1 


21. 1 


28.1 


35.2 


42.2 


49.2 


56.3 


63.3 


15.9 


I 


2.25 


7.50 


15.0 


22.5 


30.0 


37.5 


45.0 


52.5 


60.0 


67.5 


14.9 



inches. 
5/16 

H 
7/16 

9/x6 

"As 
H 

'3/16 

H 

«s/x6 



sq. in. 

.594 

.742 

.891 
1.04 
1. 19 

1-34 
1.48 

1.67 

1.78 



.08 



1-9 

2.2 
2.3 



lbs. 

1.98 
2.47 
2.97 
3.46 
396 

4.45 
4.95 
5-44 
5-94 

6.43 

6.93 
7.42 

7.92 



lbs. 

3.96 
4.95 
5.94 
6.93 
7.92 

8.91 
9.90 
10.9 
II. 9 

12.9 

13.9 
14.8 

15.8 



lbs. 

5-94 
7.42 

8.91 

10.4 

II. 9 

134 
14.8 

16.3 

17.8 

19.5 
20.8 

22.3 

23.8 



lbs. 

7.92 
9.90 
II.9 

139 
15.8 

17.8 
19.8 
21.8 
23.8 

25-7 

27.7 
29.7 

31.7 



lbs. 

9.90 
12.4 
14.8 

'73 
19 



i 



22.3 

24.7 

27.2 

29.7 

32.2 

34-6 

37.1 
39.6 



lbs. 

II. 9 
14.8 
17.8 
20.8 
23.8 

26.7 
29.7 

32.7 
35.6 

38.6 
41.6 

44.5 
47.5 



lbs. 

13-9 

17.3 
20.8 

24.2 
27-7 

31.2 

34-6 
38.1 
41.6 

45 -o 

48.5 
5".9 
55-4 



lbs. 

15.8 
19.8 
23.8 
27.7 

3J.7 

35-6 
39.6 

43-5 
47.5 

51.5 
55-4 
59.4 
63.3 



lbs. 

17.8 
22.3 
26.7 
312 
35-6 

40.1 

43.5 
49.0 

53-4 

57.9 
62.3 

66.8 
71.3 



feet. 

56.6 

45.3 
37.7 
32.3 
28.3 

25.2 
22.6 
20.6 
18.9 

17.4 
16.2 
15.1 
14.2 



FLAT BAR IRON. 



231 



Weight of Flat Bar Iron. 
2% INCHES Wide. 



' 










Length in ] 


Fekt. 








T *t 


Thick- 


Sbct. 
Area. 




















Len^h 
to weigh 


KESS. 
























X 


a 


3 


4 


5 


6 


7 


8 


9 


I cwt. 


iaches. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet. 


% 


.625 


2,08 


4.17 


6.25 


8.33 


10.4 


12.5 


14.6 


16.7 


18.8 


53.8 


5/16 


.781 


2.60 


5.21 


7.81 


10.4 


13.0 


15.6 


18.2 


20.8 


23.4 
28.1 


43.0 


H 


.938 


3.>3 


6.25 


9.38 


'2-5 


15.6 


18.8 


21.9 


25.0 


35.8 


7/16 


1.09 


3.65 


7.29 


10.9 


14.6 


18.2 


21.9 


25.5 


29.2 


32.8 


30.7 


'A 


1.25 


4.17 


8.33 


12.5 


16.7 


20.8 


25.0 


29.2 


33.3 


37.5 


26.9 


9/t6 


I.4I 


4.69 


9.38 


14. 1 


18.8 


23.4 


28.1 


32.8 


37-5 


42.2 


23.9 


H 


1.56 


5.21 


10.4 


15.6 


20.8 


26.0 


3».3 


36.5 


41-7 


46.9 


21.5 


"A6 


1.72 


S-73 


II.5 


17.2 


22.9 


28.6 


34.4 


40.1 


45.8 


51.6 


19.6 


H 


I.S8 


6.25 


12.5 


18.6 


25.0 


31.3 


37.5 


43.8 


50.0 


56.3 


18.0 


n/i6 


2.03 


6.77 


13.5 


20.3 


27.1 


33.8 


40.6 


47.4 


54.2 


60.9 


16.5 


'^, 


2.19 


7.29 


14.6 


21.9 


29.2 


36.5 


43.7 


51.0 


58.3 


65.7 


15.4 


iS/i6 


2-34 


7.81 


15.6 


23.4, 


31.3 


39.0 


46.9 


54.7 


62.5 


70.3 


14-3 


I 

1 


2.50 


8.33 


16.7 


25.0 


33-3 


41.7 


50.0 


58.3 


66.7 


75.0 


13.4 



2^ INCHES Wide. 



tnrbrf. 


sq. in. 

.656 


lbs. 

2.19 


lbs. 
4.38 


lbs. 

6.56 


lbs. 
8.75 


lbs. 
10.9 


lbs. 
13.1 


lbs. 
15.3 


lbs. 
17.5 


lbs. 

19.7 


feet. 

5>.2 


J4, 


5/.6 


.820 


2.73 


5-47 


8.20 


10.9 


13.7 


16.4 


19. 1 


21.9 


24.6 


41.0 


H 


.984 


3.28 


6.56 


9.84 


13.1 


16.4 


19.7 


23.0 


26.2 


29.5 


34.2 


7/16 


I.I5 


3. 8 J 


7.66 


II.5 


15.3 


19. 1 


23.0 


26.8 


30.6 


34.4 


29.3 


}i 


I.3I 


4.38 


8.75 


13.1 


17.5 


21.9 


26.3 


30.6 


3S-0 


39.4 


25.6 


,/;6 


1.48 


4.92 


9.84 


14.8 


19.7 


24.6 


29.5 


34-5 


39-4 


44.3 


22.8 


^, 


1.64 


5-47 


10.9 


16.4 


21.9 


27.3 


32.8 


38.3 


43.8 


49.2 


20.5 


"A« 


I.8I 


6.02 


12.0 


18. 1 


24.1 


30.2 


1 36.1 


42.1 


48.1 


54.1 


18.6 


H 


1.97 


6.56 


13. 1 


19.7 


26.3 


32.8 


1 39.4 


45.9 


52.5 


59.1 


17. 1 


•3/16 


2.13 


7.11 


14.2 


21.3 


28.4 


35-5 


1 42.7 


49.8 


56.9 


64.0 


15.8 


^. 


2.30 


7.66 


15-3 


23.0 


30.6 


38.3 


! 45.9 


53-6 


61.3 


68.9. 


14.7 


»s/lfi 


2.46 


8.20 


16.4 


24.6 


32.8 


41.0 


I 49.2 


57.4 


65.6 


73.8 


13.7 


I 


2.63 


8.75 


17.5 


26.3 


35.0 


43.8 


52.5 


61,3 


70.0 


78.8 


12.8 



2^ INCHES Wide. 



1 

indies. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet. 


X 


.688 


2.29 


4.58 


6.87 


9.17 


II.5 


13.8 


15.0 


18.3 


20.6 


48.9 


5/x6 


.859 


2.86 


5-73 


8.59 


II. 5 


14.3 


17.2 


20.1 


22.9 


25.8 


39.1 


H 


1.03 


3-44 


6.88 


10.3 


13.8 


17.2 


20.6 


24.1 


27.5 


309 


32.8 


7/16 


1.20 


4.01 


8.02 


12.0 


16.0 


20.1 


24.1 


28.1 


32.1 


36.1 


27.9 


}^ 


1-38 


4.58 


9.17 


13.8 


18.3 


22.9 


27.5 


32.1 


36.7 


41.3 


24.4 


9/16 


1.55 


5.16 


IO-3 


15.5 


20.6 


25.8 


30.9 


36.1 


41.3 


46.4 


21.7 


H 


1.72 


5.73 


11.5 


17.2 


22.9 


28.6 


34.4 


40.1 


45.8 


51.6 


*9-5 


"/t6 


1.89 


6.30 


12.6 


18.9 


25.2 


31.5 


37.8 


44.1 


50.4 


56.7 


17.8 


M 


2.06 


6.88 


138 


20.6 


27.5 


34.4 


41.3 


48.1 


55.0 


61.9 


16.3 


n/16 


2.23 


7.45 


14.9 


22.3 


29.8 


37.2 


44.7 


52.1 


59.6 


67.0 


15.0 


Ji 


2.41 


8.02 


16.0 


24.1 


32.1 


40.1 


48.1 


56.1 


64.2 


72.2 


14.0 


*S/x6 


2.58 


8.59 


17.2 


25.8 


34-4 


430 


51.6 


60.1 


68.8 


77.3 


13.0 


1 


2.75 


9.17 


18.3 


27.5 


36.7 


45.8 


55.0 


64.2 


73.3 


82.5 


12.2 



232 



WEIGHT OF METALS. 



Weight of Flat Bar Iron. 
2% INCHES Wide. 













Length in ] 


Feet. 










Thick- 
ness. 


Sect. 
Area. 




















Length 
to weigh 
























z 


2 


3 


4 


5 


6 


7 


8 


9 


X cwt. 


lAches. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet. 


%, 


.719 


2.40 


4.79 


7.19 


9.58 


12.0 


14.4 


16.8 


19.2 


21.6 


46.7 


5/16 


.898 


3.00 


6.cx> 


9.00 


12.0 


15.0 


18.0 


21.0 


24.0 


27.0 


37.4 


H 


1.08 


3.59 


7.19 


10.8 


14.4 


iS.o 


21.6 


25.2 


28.8 


32.3 


312 


7/16 


1.26 


4.19 


8.39 


12.6 


16.8 


21.0 


25.2 


29.4 


33-5 


37.7 


26.7 


'A 


1.44 


4-79 


9.58 


14.4 


19.2 


24.0 


28.8 


33-5 


38.3 


43.1 


23.4 


9/16 


1.62 


5-39 


10.8 


16.2 


21.6 


27.0 


32.3 


37.7 


^3-' 


48.5 


20.8 


^, 


1.80 


5-99 


12.0 


18.0 


24.0 


30-0 


36.0 


42.0 


48.0 


54,0 


18.7 


"A6 


1.98 


6.59 


13.2 


19.8 


26.4 


33.0 


40.5 


46.1 


52.7 


59.3 


17.0 


H 


2.16 


7.19 


14.4 


21.6 


28.8 


36.0 


43.1 


50.3 


57.5 


64.7 

1 


15.6 


t' 


2.34 


7.79 


^H 


23.4 


311 


39.0 


46.7 


54.5 


62.3 


70.1 


14.4 


2.52 


8.39 


16.8 


25.2 


33.5 


42.0 


503 


58.7 


67.1 


75-5 


13.4 


»5/i6 


2.70 


8.98 


18.0 


27.0 


35.9 


45.0 


53.9 


62.9 


71.9 


80.9 


12.4 


I 


2.88 


9.58 


19.2 


28.8 


38.3 


48.0 


57-5 


67.1 


76.7 


86.3 


II.7 



3 INCHES Wide. 



inches. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


1 

lbs. 


lbs. 


lbs. 


lbs. 


feet. 


% 


.750 


2.50 


5.00 


7.50 


lO.O 


12.5 


15.0 


17.5 


20.0 


22.5 


44.8 


s/i6 


.938 


3-13 


6.25 


9.38 


I2.S 


'^Z 


18.8 


21.9 


25.0 


28.1 


35.8 


H 


1. 13 


3.75 


7.50 


II-3 


15.0 


18.8 


22.5 


26.3 


30.0 


33.8 


29.9 


7/i6 


I-3I 


438 


8.75 


131 


17.5 


21.9 


26.3 


30.6 


350 


39.4 


25.6 


'A 


1.50 


5.00 


lO.O 


15.0 


20.0 


25.0 


30.0 


35.0 


40.0 


45.0 


22.4 


9/16 


1.69 


5-63 


"3 


16.9 


22.5 


28.2 


33.8 


39.4 


450 


50.6 


.19.9 


H 


1.88 


6.25 


12.5 


18.8 


25.0 


31-3 


37.5 


43.8 


50,0 


56.3 


17.9 


r 


2.06 


6.88 


13.8 


20.6 


27.5 


34.4 


41.3 


48.1 


55.0 


61.9 


16.3 


2.25 


7.50 


15.0 


22.5 


30.0 


37-5 


45.0 


52.5 


60.0 


67.5 


14.9 


T 


2.44 


8.13 


16.3 


24.4 


32.5 


40.7 


48.8 


56.9 


65.0 


73-' 


13.8 


2.63 


8.75 


'7-5 


26.3 


35.0 


43.8 


52.5 


61.3 


70.0 


78.8 


12.8 


^5/16 1 


2.81 


9.38 


18.8 


28.1 


37-5 


46.9 


56.3 


65.6 


75.0 


84.4 


12.0 


I 1 

1 


3-0O 


10.0 


20.0 


300 


40.0 


50.0 


60.0 


70.0 


80.0 


90.0 


II. 2 



3X inches Wide. 



inches. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


fecL 


X 


.813 


2.71 


5.42 


8.13 


10.8 


13.6 


16.3 


19.0 


21.7 


24.4 


41.3 


5/16 


1.02 


3.39 


6.77 


10.2 


13.5 


16.9 


20.3 


23.7 


27.1 


30.5 


33.1 


H 


1.22 


4.06 


8.13 


12.2 


16.3 


20.3 


24.4 


28.4 


32.5 


36.6 


27.5 


7/16 


1.42 


4.74 


9.48 


14.2 


19.0 


237 


28.4 


33.2 


37.9 


4?/Z 


23.6 


yi 


1.63 


5.42 


10.8 


16.3 


21.7 


27.1 


32.5 


37.9 


43-3 


48.8 


20.7 


9/.« 


1.83 


6.09 


12.2 


18.3 


24.4 


305 


36.6 


42.7 


48.7 


54.8 


18.4 


H 


2.03 


6.77 


13.5 


20.3 


27.1 


33.9 


40.6 


47.4 


54.2 


60.9 


16.5 


r 


2.23 


7-45 


14.9 


22.3 


29.8 


37.2 


44.7 


52.1 


59.6 


67.0 


15.0 


2.44 


8.13 


16.3 


24.4 


32.5 


40.6 


48.8 


56.9 


65.0 


731 


13.7 


r 


2.64 


8.80 


17.6 


26.4 


35.2 


44.0 


52.8 


61.6 


70.4 


79.2 


12.7 


2.84 


9.48 


19.0 


28.4 


37.9 


47.4 


56.9 


66.4 


75.8 


85.3 


II.8 


»5/i6 


3.05 


10.2 


20.3 


30. 5 


40.6 


50.8 


60.9 


71. 1 


81.2 


91.4 


II.O 


I 


325 


10.8 


21.7 


32. 5 


43.3 


54.2 


65.0 


75.8 


86.7 


97.5 


10.3 



FLAT BAR IRON. 



233 



Weight of Flat Bar Iron. 
S}4 INCHES Wide. 



1 










Lbngth in 


Fbbt. 










Thick- 1 


Sect. 
Area. 




















Lens|th 
to weigh 


















 






I 


a 


3 


4 


5 


6 


7 


8 


9 


I CWL 


1 inches. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet. 


><■ 


.875 


2.92 


5.83 


8.75 


II.7 


14.6 


17.5 


2G.4 


233 


26. J 


38.4 


s/16 


1.09 


3-^5 


7.29 


10.9 


14.6 


18.2 


21.9 


25.5 


29.2 


32.8 


30.7 


H 


I-3I 


4.38 


8.75 


13. 1 


17.5 


21.9 


26.3 


30.6 


35.0 


39.4 


25.6 


7/16 


1.53 


5.10 


10.2 


15.3 


20.4 


25.5 


30.6 


35-Z 


40.8 


45-9 


21.9 


}i 


1.75 


5.83 


II. 7 


17.5 


22.3 


29.2 


35.0 


40.8 


46.7 


52.5 


19.2 


S/.6 ' 


1.97 


6.56 


13. 1 


19.7 


26.3 


32.8 


39.4 


45-9 


55-5 


5^1 


17. 1 


H 


2.19 


7.29 


14.6 


21.9 


29.2 


36.5 


43.7 


51.0 


58.3 


65.6 


15-4 


T 


2.41 


8.02 


16.0 


24.1 


32.1 


40.1 


48.1 


56.1 


64.2 


72.2 


14.0 


2.63 


8.75 


«7.5 


26.3 


350 


43.8 


52.5 


61.3 


70.0 


78.8 


12.8 


'3/16 


2.84 


9.48 


19.0 


28.4 


37.9 


47.4 


56.9 


66.4 


75.8 


85.3 


11.9 


H 


3.06 


10.2 


20.4 


306 


40.8 


51.0 


61.2 


71.5 


81.6 


91.9 


II. 


«5/i6 


3.28 


10.9 


21.9 


32.8 


43.8 


54.7 


65.6 


76.6 


87.5 


98.4 


10.2 


I 


' 3.50 


II.7 


23.3 


35.0 


46.7 


58.3 


70.0 


81.7 


93.3 


105.0 


9.60 



3j^ INCHES Wide. 



t 



inches. 

5/16 

H 

7/16 

9/16 
H 

"1x6 

H 

«j/i6 

»5/i6 

I 



sq. m. 

.938 
I.I7 
I.4I 
1.64 

1.88 

2. II 

2.34 

2.58 

2.81 

3.05 
3-28 
3.52 
3-75 



lbs. 


lbs. 


3.13 
3.91 
4.69 

6.25 


6.25 
7.81 

9.38 

10.9 

12.5 


7.03 

7.81 

8.59 
9.38 


14. 1 

15.6 

17.2 
18.8 


10.2 


20.3 


10.9 

II.7 
12.5 


21.9 

23.4 

25.0 



lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet. 


12.5 


15.6 


18.8 


21.9 


25.0 


28.1 


35-^ 


15.6 


19.5 


23.4 


27.3 

32.8 


3^.3 


35.2 


28.7 


18.8 


234 


28.1 


37.5 


42.2 


23.9 


21.9 


27.3 


32.8 


3^-2 


43.7 


49.2 


20.5 


25.0 


3».3 


37.5 


43.8 


50.0 


56.3 


17.9 


28.1 


35.3 


42.2 


49.2 


56.3 


63.3 


15.9 


3>.2 


39.1 


46.9 


54.7 


62.5 


70.3 


'4.3 


34.4 


43.0 


51.6 


60.2 


68.8 


27-3 


13.0 


37.5 


46.9 


56.3 


65.6 


75.0 


84.4 


12.0 


40.6 


50.8 


60.9 


7I.I 


81.3 


91.4 


II.O 


43.8 


54.7 


65.6 


76.6 


87.5 


98.4 


10.2 


46.9 


58.6 


70.3 


82.0 


93-7 


105.5 


2-5^ 


50.0 


62.5 


75.0 


87.5 


100. 


112.5 


8.96 



4 INCHES Wide. 



inchrt. 


iq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lU. 


lbs. 


lbs. 


feet. 


'^ 1 


I.OO 


3.33 


6.67 


10. 


13.3 


16.7 


20.0 


23.3 


26.7 


30.0 


33.6 


5/16 ! 
H 


1.25 


4.17 


8.33 


12.5 


16.7 


20.8 


25.0 


29.2 


33.3 


37.5 


26.9 


1.50 


5.00 


lO.O 


15.0 


20.0 


25.0 


30.0 


35.0 


40.0 


45.0 


22.4 


7/16 


1-75 


5.83 


II.7 


17.5 


23-3 


29.2 


35.0 


40.8 


46.7 


52.5 


19.2 


H 1 


2.00 


6.67 


13.3 


20.0 


26.7 


33.3 


40.0 


46.7 


53-3 


60.0 


i6.8 


9/16 1 


2.25 


7.50 


15.0 


22.5 


30.0 


37.5 


45.0 


5;- 5 


60.0 


67.5 


14.9 


H 


2.50 


8.33 


16.7 


25.0 


33-3 


41.7 


50.0 


58.3 


66.7 


75.0 


13.4 


"/«6 


2.75 


9.17 


18.3 


27.5 


36.7 


45.8 


55.0 


64.2 


73.3 


82.5 


12.2 


1 H ' 


' 300 


lO.O 


20.0 


30.0 


40.0 


50.0 


60.0 


70.0 


80.0 


90.0 


II. 2 


13/.6 1 


1 3.25 


10.8 


21.7 


32.5 


43-3 


54.2 


65.0 


Z5-» 


86.7 


97.5 


10.3 


^, 


3.50 


II.7 


23.3 


35.0 


46.7 


58.4 


70.0 


81.7 


93.3 


105.0 


9.60 


' «5/.6 


, 3-75 


12.5 


25.0 


37.5 


50.0 


62.5 


75.0 


87.5 


lOO.O 


II2.5 


8.96 


I 


4.00 


13.3 


26.7 


40.0 


53.3 


66.7 


80.0 


93.3 


106.7 


120.0 


8.40 



234 



WEIGHT OF METALS. 



Weight of Flat Bar Iron. 
4X INCHES Wide. 











Length in 


Feet. 










Thick- 


Sect. 
Area. 


















Len^h 
to weigh 


ness. 
























I 


2 


3 


4 


5 


6 


7 


8 


9 


I OWL 


inches. 


1 
sq. ixu 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet 


V 


1.06 


3^54 


7.08 


10.6 


14.2 


17.7 


21.3 


24.8 


28.3 


31.9 


31.6 


5/16 


1.33 


4.43 


8.85 


13.3 


17.7 


22.1 


26.6 


3I-0 


35.4 


39.8 


25.3 


H 


^•59 


5.31 


10.6 


»5-9 


21.3 


26.6 


31.9 


37.2 


42.5 


47.8 


21. 1 


7/16 


1.85 


6.20 


12.4 


18.6 


24.8 


31.0 


37.2 


43-4 


49.6 


55.8 


18. 1 


^ 


2.13 


7.08 


14.2 


21.3 


28.3 


35.4 


42.5 


49.6 


56.7 


63.8 


15.8 


9/16 


2.39 


ni 


15.9 


23.9 


31.9 


39.8 


47.8 


55.8 


63.7 


71.7 


14. 1 1 


H, 


2.66 


17.7 


26.6 


35.4 


44.3 


53-1 


62.0 


70.8 


79.7 


12.7 


t' 


2.92 


9.74 


19.5 


29.2 


39.0 


48.7 


58.4 


68.2 


77.9 


87.7 


II.5 


3-19 


10.6 


21.3 


31.9 


42.5 


53-1 


63.8 


74-4 


85.0 


95.6 


10.5 


«3/i6 


3.45 


11.5 


23.0 


34.5 


46.0 


57.6 


69.1 


80.6 


92.1 


103.6 


9-9 


n 


3.72 


12.4 


24.8 


37.2 


49.6 


62.0 


74.4 


86.8 


99.2 


III. 6 


9.0 


^5/16 


3.98 


13.3 


26.6 


39.« 


53.1 


66.4 


79.7 


93-0 


106.2 


"95 


8.4 


I 


4.25 


14.2 


28.3 


42.5 


56.7 


70.8 


85.0 


99.2 


"33 


127.5 1 7.9 



4}4 iN'CHEs Wide. 



inches. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. i 


lbs. 


lbs. 


lbs. 


lbs. 


feet 


^ 


I.I3 


3.75 


7-5, 


"•3 


15.0 


18.8 


22.5 


26.3 


30-0 


33.8 


29.9 


5/16 


I.4I 


4.69 


9.38 


14. 1 


18.8 


\l\ 


28.1 


32.8 


37.5 


42.2 


23.9 


^ 


1.69 


5.63 


"3 


16.9 


22.5 


33.8 


39-4 


45.0 


50.6 


19.9 


7/16 


1.97 


6.56 


131 


19.7 


26.3 


32.8 , 


39-4 


45-9 


52.5 


59.1 


17. 1 


H 


2.25 


7.50 


15.0 


22.5 


30.0 


37.5 1 


45.0 


52.5 


60.0 


67.5 


14.9 


9/16 


2.53 


8.44 


16.9 


25.3 


33.8 


42.2 


50.6 


59.1 


67.5 


75.9 


13-3 


H 


2.81 


9.38 


18.8 


28.1 


37.5 


46.9 , 


56.3 


65.6 


75.0 


84.4 


12.0 


r 


3.09 


10.3 


20.6 

• 


30.9 


41.3 


51.6 


61.9 


72.2 


82.5 


92.8 


10.9 


3-38 


"3 


22.5 


33.8 


45.0 


56.3 


67.5 


78.8 


90.0 


IOI.3 


9-95 


t' 


3.66 


12.2 


24.4 


36.6 


48.8 


60.9 ' 


721 


85.3 


97.5 


109.7 


9.19 


3-94 


13.1 


•26.3 


39-4 


52.5 


65.6 


78.8 


91.9 


105.0 


118. 1 


».53 


»s/i6 


4.22 


14. 1 


28.1 


42.2 


56.3 


70.3 


84.4 


98.4 


II2.5 


126.6 


7.96 


I 


4.50 


15.0 


30.0 


45.0 


60.0 


75.0 


90.0 


105.0 


120.0 


135.0 


7.46 



^}( INCHES Wide, 



inches. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet 


% 


1. 19 


3.96 


7.92 


II.9 


15.8 


19.8 


23.8 


27.7 


31.7 


35.6 


28.3 


she 


1.48 


4.95 


9.90 


14.8 


19.8 


24.8 


29.7 


34.6 


39.6 


44.4 


22.6 


H 


1.78 


5-94 


11.9 


17.8 


23.8 


29.7 


35-6 


41.6 


47.5 


53.4 


18.9 


7/16 


2.08 


6.93 


'3-2 


20.8 


27.7 


34-7 


41.6 


48.5 


55-4 


62.3 


16.2 


^ 


2.38 


7.92 


15.8 


23.8 


31.7 


39.6 


47.5 


55.4 


63.3 


71.3 


14.2 


9/.6 


2.67 


8.91 


17.8 


26.7 


35.6 


44.6 


53-4 


62.3 


71.3 


80.2 


12.6 


^, 


2.97 


9.90 


19.8 


29.7 


39.6 


49-5 


1 59-4 


69-3 


79.2 


89.1 


"3 


r 


3.27 


10.9 


"S 


32.7 


43.5 


54.5 


' 65.3 


76.2 


87.1 


98.0 


10.3 


3.56 


1 1.9 


23.8 


35.6 


47. 5 


59.4 


71.3 


83.1 


95.0 


106.9 


9-4 


«3/i6 


3.86 


12.9 


25.7 


38.6 


51.5 


64.3 


77.2 


90.1 


102.9 


II5.8 


8.7 


^> 


4.16 


13-9 


27.7 


41.6 


55.4 


69.3 


83.1 


97.0 


no. 8 


124.7 


8.1 


'5/16 


4.45 


14.8 


29.7 


44.5 


59.4 


74.2 


89.1 


103.Q 


118.8 


133-6 


7.5 


I 


4.75 


15.8 


31-7 


47.5 


633 


79.2 


95.0 


1 10.8 


126.7 


142.5 


7.1 



FLAT BAR IRON. 



235 



Weight of Flat Bar Iron. 
5 INCHES Wide. 













Length in ] 


Feet. 










Thick- 
ness. 


Sect. 

ASEA. 




















Len^h 
to weigh 
























z 


2 


3 


4 


5 


6 


7 


8 


9 


I cwt. 


ipches. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet. 


u 


1.25 


4.17 


8.33 


12.5 


16.7 
20.8 


20.9 


25.0 


29.2 


33.3 


37.5 


26.9 


S/>6 


l!88 


5.21 


10.4 


15.6 


26.1 


3>.3 


36. J 
43.8 


41.7 


46.9 


21.5 


H 


6.25 


12.5 


18.8 


25.0 


3>.3 


37-5 


50.0 


56.3 


17.9 


7/16 


2.19 


7.29 


14.6 


21.9 


29.2 


36.5 


43.8 


5i-° 


58.3 


65.6 


15.4 


a 


2.50 


8-33 


16.7 


25.0 


33.3 


41.7 


50.0 


58.3 


66.7 


75-0 


13.4 


9/16 


2.81 


9.38 


18.8 


28.1 


37.5 


46.9 


56.3 


65.6 


75.0 


84.4 


12.0 


><, 


3.13 


10.4 


20.8 


313 


41.7 


52.1 


62.5 


72.9 


83.3 


93-8 


10.8 


r 


3-44 


11.5 


22.9 


34.4 


45.8 


57.3 


68.8 


80.2 


91.7 


103. 1 


9-77 


3.75 


12.5 


25.0 


37.5 


50.0 


62.5 


75.0 


87.5 


100. 


112.5 


8.96 


't 


4.06 


13.5 


27.1 


40.6 


54.2 


67.7 


81.3 


94.8 


108.3 


121. 9 


8.27 


4.38 


14.6 


29.2 


43.8 


58.3 


72.9 


87.5 


102. 1 


II6.7 


'31.3 


7.68 


«5/i6 


4.69 


15.6 


31.3 


46.9 


62.5 


78.1 


93.8 


109.4 


125.0 


140.6 


7.17 


I 


5.00 


16.7 


33.3 


50.0 


66.7 


83.3 


lOO.O 


116. 7 


133.3 


150.0 


6.72 



5^ INCHES Wide. 



inchfs 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet 


V 


I-3I 


4.38 


8.75 


13-1 


17.5 


21.9 


26.3 
32.8 


30.6 


35.0 


39-4 


25.6 


s/16 


1.64 


§•47 


10.9 


16.4 


21.9 


27.3 


38.3 


43.8 


49.2 


20.5 


H 


1.97 


6.56 


"3-1 


19.7 


26.3 


32.8 


39-4 


1 45-9 


52.5 


59.1 


I7.I 


7/16 


2.30 


7.66 


15.3 


23.0 


30.6 


38.3 


45-9 


53.6 


61.3 


68.9 


14.6 


>i 


2.63 


8.75 


17.5 


26.3 


35-0 


43-8 


52.5 


61.3 


70.0 


78.8 


12.8 


9/16 


2.95 


9.84 


19.7 


29.5 


39.4 
43.8 


49.2 


59.1 


68.9 


78.8 


88.6 


II.4 


H 


3.28 


10.9 


21.9 


32.8 


54-7 


65.6 


76.6 


87.5 


98.4 


I0.3 


r 


3.61 


12.0 


24.1 


36.1 


48.1 


60.2 


72.2 


84.2 


96.3 


108.3 


9.31 


3.94 


13. 1 


26.3 


39.4 


52.5 


65.6 


78.8 


91.9 


105.0 


118.1 


8.55 


»3/i6 


4.27 


14.2 


28.4 


42.7 


56.9 


71. 1 


85.3 


99.5 


113.7 


128.0 


7.88 


H 


4.59 


»5.3 


30.6 


45.9 


61.3 


76.6 


91.9 


107.2 


122.5 


137.8 


7-3* 


^s/i6 


4.92 


16.4 


32.8 


49.2 


65.6 


82.0 


98.4 


114.8 


131. 3 


147.7 


6.S3 


I 


5.25 


17.5 


35.0 


52.5 


70.0 


87.5 


105.0 


122.5 


140.0 


157.S 


6.40 



$}4 INCHES Wide. 



indies. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet. 


'X 


1.38 


4.58 


9.17 


13.8 


18.3 


22.9 


27.5 


32.1 


36.7 


41.3 


24.5 


S/t6 


1.72 


m 


11. g 


17.2 


22.9 


28.6 


34.4 


40.1 


45.8 


51.6 


'95 


H 


2.06 


13.8 


20.6 


27.5 


34.4 


4i-3 


48.1 


55.0 


61.9 


16.4 


7/16 


2.41 


8.02 


16.0 


24.1 


32.1 


40.1 


48.1 


56.1 


64.2 


72.2 


14.0 


>i 


2.75 


9-17 


18.3 


27.5 


36.7 


45.8 


55-0 


64.2 


73.3 


82.5 


12.2 


9/16 


3.09 


W.3 


20.6 


30.9 


41.3 


51.6 


61.9 


72.2 


82.5 


92.8 


10.9 


H 


3.44 


11.5 


22.9 


34.4 


45.8 


57.3 


68.8 


80.2 


91.7 


103. 1 


9.77 


"ft6 


3.78 


12.6 


25.2 


37.8 


50.4 


63.0 

68.8 


75.6 


88.2 


100.8 


113.4 


8.88 


H 


4.13 


13.8 


27.5 


41.3 


55.0 


82.5 


96.3 


IIO.O 


123.8 


8.14 


T 


1 4-47 


14.9 


29.8 


44.7 


59.6 


74.5 


89.4 


104.3 


119. 2 


134. 1 


7.52 


4.81 


16.0 


3^.1 


48.1 


^l 


80.2 


96.3 


H2.3 


128.3 


144.4 


6.98 


»5/i6 


5.16 


17.2 


34.4 


51.6 


68.8 


85.9 


103. 1 


120.3 


137.5 


154.7 


6.52 


' 


5-50 


"8.3 


36.7 


55.0 


73-3 


91.6 


IIO.O 

1 


128.4 


146.7 


165.0 


6.11 



236 



WEIGHT OF METALS. 



Weight of Flat Bar Iron. 
$}( INCHES Wide. 













Lbngth in Fbbt. 








T *.!_ 


Thick- 
ness. 


Sbct, 
Arba. 




















Length 
to weigh 
























z 


2 


3 


4 


5 


6 


7 . 


8 


9 


X cwt. 


inches. 


1 
sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet. 


m: 


1.44 


4.79 


9.58 


14.4 


19.2 


24.0 


28.8 


33.5 


38.3 


43-1 


23.4 


s/16 


1.80 


5-99 


12.0 


i8.o 


24.0 


30.0 


35.9 


41.9 


47.9 


53.9 


18.7 


H 


2.16 


7.19 


14.4 


21.6 


28.8 


35.9 


43.1 


50.3 


57.5 


64.7 


15.6 


7/j« 


2.52 


8.39 


16.8 


25.2 


33.5 
38.3 


41.9 


50.3 


58.7 


67.1 


75-5 


134 


H 


2.88 


9.58 


19.2 


28.8 


47.9 


57.5 


67.1 


76.7 


86.3 


II.7 


9/«« 


3.23 


10.8 


21.6 


32.3 


43.' 
48.0 


53.9 


64.7 


75.5 


86.2 


97.0 


10.4 


H 


3.59 


12.0 


24.0 


36.0 


60.0 


71.9 


83.9 


95.8 


107.8 


9-35 


r 


3.95 


13.2 


^t-i 


39.5 


52.7 


65.9 


79.1 


92.2 


105.4 


1 18.6 


8.50 


4.31 


14.4 


28.8 


43.1 


57.5 


71.9 


86.3 


100.6 


115.0 


129.4 


7.79 


r 


4.67 


15.6 


31.2 


46.7 


62.3 


77.9 


93-4 


109.0 


124.6 


140.2 


7.19 


5.03 


16.8 


33.5 


50.3 


67.0 


83.9 


ic».7 


1 17.4 


134.2 


150.9 


6.68 


'5/x6 


5.39 


18.0 


35.9 


53-9 


71.9 


89.8 


107.8 


125.8 


H3-7 


161. 7 


6.22 


I 


5.75 


19.2 


38.3 


57.5 


76.7 


95.8 


115.0 


"34.2 


153.3 


172.5 


5.83 



6 INCHES Wide. 



inches. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet. 


H 


1.50 


5.00 


lO.O 


15.0 


20.0 


25.0 


30.0 


35.0 


40.0 


45.0 


22.4 


s/x6 


1.88 


6.25 


12.5 


18.8 


25.0 


31.8 


37.5 


43.8 


50.0 


56.3 


17.9 


H 


2.25 


7.50 


15.0 


22.5 


30.0 


37-5 


45.0 


52.5 


60.0 


67.5 


14.9 


7/16 


2.63 


8.75 


17.5 


26.3 


350 


43.8 


52.5 


61.3 


70.0 
80.0 


78.8 


12.8 


H 


3.00 


10.0 


20.0 


30.0 


40.0 


50.0 


60.0 


70.0 


90.0 


II. 2 


9/x6 


3.38 


"3 


22.5 


33.8 


45.0 


56.3 


67.5 


78.8 


90.0 


101.3 


lO.O 


H 


3-75 


12.5 


25.0 


37.5 


50.0 


62.5 


75.0 


87.5 


100.0 


112.5 


8.96 


r 


4.>3 
4.50 


13.8 
15.0 


27.5 
30.0 


41-3 
450 


60.0 


68.8 
75.0 


82.5 

90.0 


96.3 
105.0 


HO.O 

120.0 


123.7 
135.0 


8.15 
7.47 


t 


4.88 


16.3 


32.5 


48.8 


65.0 


81.3 


97.5 


"37 


1300 


146.3 


6.90 


5.25 


'7-5 
18.8 


35.0 


52.5 


70.0 


87.$ 


105.0 


122.5 


140.0 


157.5 


6.40 


>5/i6 


5.63 


37.5 


56.3 


75.0 


93.8 


II2.5 


131.3 


150.0 


168.7 


5-97 


I 


6.00 


20.0 


40.0 


60.0 


80.0 


lOO.O 


120.0 


140.0 


160.0 


180.0 


S.60 



6% INCHES Wide. 



inches. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet. 


% 


1.63 


5-42 


10.8 


16.3 


21.7 


27.2 


32.5 


37.9 


43.3 


49.0 


20.7 


5/x6 


2.03 


6.77 


13.5 


20.3 


27.1 


33-9 


40.6 


47.4 


54.2 


60.9 


16.5 


H 


2.44 


8.13 


16.3 


24.4 


32.5 


40.6 


48.8 


56.9 


65.0 


73.1 


7/x6 


2. 84 


9.47 


18.9 


2S.4 


37.9 


47.4 


56.8 


66.3 


75.8 


85.2 


14.8 


}i 


3.25 


10.8 


21.7 


32.5 


43.3 


54.2 


65.0 


75.8 


86.7 


97.5 


10.3 


9/x6 


3.66 


12.2 


24.4 


36.6 


48.8 


60.9 1 


73- ' 


85.3 


97-5 


109.7 


9.20 


>^. 


4.06 


13.5 


27.1 


40.6 


54.2 


67.7 


81.3 


94.8 


108.3 


121.9 


8.27 


"/i6 


4-47 


14.9 


29.8 


44.7 


59.6 


74.5 


89.4 


104.3 
1 13.8 


119.2 


134.1 


6.89 


H 


4.98 


16.3 


32.5 


48.8 


65.0 


81.3 


97.5 


130.0 


146.3 


'3/x6 


5.28 


17.6 


35-2 


52.8 


70.4 


88.0 


105.6 


123.2 


140.8 


158.4 


6.36 


H 


5.68 


19.0 


37-9 


56.9 


75.8 
81.3 


94.8 


1 13.8 


132.7 


151.7 


170.6 


5-91 


'5/x6 


6.09 


20.3 


40.6 


60.9 


101.6 


121.9 


142.8 


162.5 


182.8 


5.51 


I 


6.50 


21.7 


43.3 


65.0 


86.7 


108.3 


130.0 

1 


I5I.7 


173-3 


195.0 


5.29 



FLAT BAR IRON. 



237 



Weight of Flat Bar Iron. 











7 


INXHES Wide. 




















Length in Feet. 






Thick- 
ness. 


Sect. 
Area. 














Length 
to weigh 
























X 


2 


3 


4 


5 


6 


7 


8 


9 


I CWL 


* 

inches. 


sq. m. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet. 


X 


1.75 


583 


II.7 


17.S 


23.3 


29.2 


35.0 


40.8 


46.7 


52.5 


19.2 


s/i6 


2.19 


7.29 


14.6 


21.9 


29.2 


3^-5 


43.8 


51.0 


58.3 


65.6 


15.4 


H 


2.63 


8.75 


17.5 


26.3 


35-2 


43-8 


52.5 


61.3 


70.0 


78.8 


12.8 


I ?/«« 


3-o6 


10.2 


20.4 


30.6 


40.8 


5?/° 


61.3 


7^-5 


81.7 


91.9 


II.O 


1 H 


3.50 


II.7 


23.3 


350 


'46.7 


58.3 


70.0 


81.7 


93-3 


105.0 


9.60 


9/16 


3-94 


13. 1 


26.3 


39.4 


5;^- 5 


65.6 


78.8 


91.9 


105.0 


I18.I 


8.53 


H 


4.38 


14.6 


29.2 


43.8 


58.3 


72.9 


87.5 


102. 1 


1 16. 7 


131.3 


7.68 


r 


4-81 
5.25 


16.0 
17.5 


32.1 
35.0 


48. 1 
52.5 


64.2 
70.0 


80.2 
87.5 


96.3 
105.0 


112.3 
122.5 


128.3 
140.0 


144-4 
157.5 


6.98 

6.40 


«3/i6 


5.69 


19.0 


37-9 


56.9 


75.8 


95.0 


II3.8 


132.7 


151.7 


170.6 


591 


H 


6.13 


20.4 


40.8 


61.3 


81.7 


102. 1 


122.5 


142.9 


163.3 


183.8 


5-49 


«5/i6 


6.56 


21.9 


43.8 


65.6 


87.5 


109.4 


131-3 


'53- » 


175.0 


196.9 


5.12 


I 


7.00 


23.3 


46.7 


70.0 


93.3 


1 16. 7 


140.0 


163.3 


186.7 


210.0 


4.80 



y}^ INCHES Wide. 



baches. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet. 


X 


1.88 


6.25 


12.5 


18.8 


25.0 


31.3 


37-5 


43.8 


50.0 


56.3 


17.9 


s/16 


2.34 


7.81 


15.6 


23.4 


31.3 


39. > 


46.9 


54.7 


62.5 


70.3 


14-3 


H 


i 2.81 


9.38 


18.8 


28.1 


37-5 


46.9 


56.3 


65.6 


75.0 


84.4 


II.9 


7/16 


3-28 


10.9 


21.9 


32.8 


43.8 


54.7 


65.6 


76.6 


87.5 


98.4 


10.2 


>i 


3.75 


12.5 


25.0 


37.5 


50.0 


62.5 


75.0 


87.5 


100. 


H2.5 


8.96 


9/16 


4.22 


14. 1 


28.1 


42.2 


56.3 


70.3 


84.4 


98.4 


II2.5 


126.6 


7.96 


H 


4.69 


15.6 


3Jf.3 


46.9 


62.5 


78.1 


93.8 


109.4 


125.0 


140.6 


7.17 


"/16 


5.16 


"^^'l 


34-4 


51.6 


68.8 


85.9 


103. 1 


120.3 


137.5 


154.7 


6.52 


H 


5.63 


18.8 


37.5 


56.3 


75.0 


93.8 


112.5 


131-3 


150.0 


168.8 


5.97 


T 


6.09 


20.3 


40.6 


60.9 


81.3 


I0I.6 


121. 9 


142.2 


162.5 


182.8 


5-51 


6.56 


21.9 


43.8 


65.6 


87.5 


109.4 


131.3 


153.1 


175.0 


196.9 


5.12 


»5/i6 


7-03 


23.4 


46.9 


70.3 


93.8 


117.2 


140.6 


164. 1 


187.5 


210.9 


4.78 


I 


7.50 


25.0 


50.0 


75.0 


100. 


125.0 


150.0 


175.0 


200.0 


225.0 


4.48 



8 INCHES Wide. 



inches. 


I sq. in. 


lbs. 


lbs. 


lbs. 


lbs 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet. 


X 


1 
2.00 


6.67 


13.3 


20.0 


26.7 


33-3 


40.0 


46.7 


53-3 


60.0 


16.8 


s/16 


2.50 


8.33 


16.7 


25.0 


33.3 


41.7 


50.0 


58.3 


66.7 


75.0 


13.4 


H 


' 300 


10. 


20.0 


30.0 


40.0 


50.0 


60.0 


70.0 


80.0 


90.0 


II. 2 


7/16 


3-50 


II.7 


233 


35.0 


46.7 


58.3 


70.0 


81.7 


93.3 


105.0 


0.60 


1 ^ 


4.00 


13.3 


26.7 


40.0 


53-3 


66.7 


80.0 


93-3 


106.7 


120.0 


8.40 


9/16 


4.50 


15.0 


30.0 


45.0 


60.0 


75.0 


90.0 


105.0 


120.0 


135.0 


7.47 


^, 


5.00 


16.7 


33.3 


co.o 


66.7 


83.3 


lOO.O 


116.7 


133.3 


150.0 


6.72 


t' 


5.50 


18.3 


36.7 


55.0 


73.3 


91.7 


IIO.O 


128.3 


146.7 


165.0 


6.11 


6.00 


20.0 


40.0 


60.0 


80.0 


100.0 


120.0 


140.0 


160.0 


180.0 


5.60 


»3/i6 


6.50 


21.7 


43.3 


65.0 


86.7 


108.3 


130.0 


151. 7 


173.3 


195.0 


5-i7 


», 


7.00 


23-3 


46.7 


70.0 


93-3 


116.7 


140.0 


163.3 


186.7 


2IOkO 


4.80 


^5fi6 


7.50 


25.0 


50.0 


75.0 


100.0 


125.0 


150.0 


175.0 


200.0 


225.0 


4.48 


1 


8.00 


26.7 


53.3 


80.0 


106.7 


133.3 


160.0 


186.7 


213.3 


240.0 


4.20 



238 



WEIGHT OF METALS. 



Weight of Flat Bar Iron. 
9 INCHES Wide. 



Thicic- 

NESS. 


Sect. 
Area. 


Length in Fbbt. 


Length 

to weigh 

I cwt. 


z 


1 


3 


4 


5 


6 


7 


8 


9 


inches. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet. 


5/16 

H 
7/16 


2.25 
2.81 

3.38 

3-94 
4,50 


7-50 
9.38 
"3 

13. 1 
15.0 


15.0 

18.8 

22.5 
26.3 
30.0 


22.5 
28.1 
33-8 
39-4 
45-0 


30.0 

37-5 
45.0 

60.0 


37.5 
46.9 

tl 
65.6 

75-0 


45.0 
56.3 

Hi 
78.8 

• 90.0 


52.5 
65.6 

78.8 
91.9 

105.0 


60.0 

75.0 

90.0 

105.0 

120.0 


67.5 

84.4 
IOI.3 

118. 1 
135.0 


14.9 
II.9 
10.0 

8.53 
7.47 


9/16 


5.06 

5.63 
6.19 

6.75 


16.9 
18.8 
20.6 
22.5 


33.8 

37-5 

41-3 
45.0 


50.6 

56.3 
61.9 

67.5 


67.5 
75.0 

82.5 

90.0 


84.4 

93-8 

103. 1 

112.5 


101.3 
112.5 
123.8 

135-0 


118. 1 

I3I.3 
144.4 

157.5 


1350 
150.0 
165.0 
180.0 


151.9 
168.8 
185.6 
202.5 


6.64 

5-97 

5-43 
4.98 


»5/i6 

I 


8.44 
9.00 


24.4 
26.3 
28.1 
30.0 


48.8 

56.3 
60.0 


73.1 
78.8 

84.4 
90.0 


97-5 
105.0 

112.5 

120.0 


121. 9 

i3>-3 
140.6 

150.0 


146.3 
104.5 

168.8 

180.0 


170.6 

183.8 

196.9 

210.0 


195.0 
210.0 
225.0 
240.0 


219,4 

236.3 

253-1 
270.0 


4.59 
4.26 

3.98 

3.73 



10 INCHES Wide. 



inches. ' 

1 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet. 


^ 


2.50 


8.33 


16.7 


25.0 


33-3 


41.7 


50.0 


58.3 


66.7 


75.0 


13-4 


S/.6 


3.13 


10.4 


20.8 


31.3 


41.7 


52.1 


62.5 


72.9 


83.3 


93.8 


10.7 


^ 


3.75 


12.5 


25.0 


37-5 


50.0 


62.5 


75.0 


87-5 


100.0 


II2.5 


8.96 


7/16 


4-38 


14.6 


29.2 


43-8 


58.3 


72.9 


87-5 


102. 1 


116.7 


I3I-3 


7.68 


}i 


5.00 


16.7 


33-3 


50.0 


66.7 


83.3 


lOO.O 


116.7 


133.3 


150.0 


6.72 


9/16 


5.63 


18.8 


37.5 


56.3 


75.0 


93.8 


II2.5 


13I-3 


150.0 


168.8 


5-97 


H 


6.25 


20.8 


^^■7 


62.5 


83-3 


104.2 


125.0 


145.8 


166.7 


187.5 


4^89 


r 


6.88 


22.9 


45.8 


68.8 


91-7 


1 14.6 


137-5 


160.4 


183-3 


206.3 


7.50 


25.0 


50.0 


75.0 


100. 


125.0 


150.0 


175.0 


200.0 


225.0 


4.48 


»3/i6 


8.13 


27.1 


54.2 


81.3 


108.3 


135-4 


162.5 


189.6 


216.7 


243.8 


4.14 


'^. 


8.75 


29.2 


58.3 


87.5 
93-8 


1 16. 7 


145.8 


175.0 


^^'i 


233-3 


262.5 


3.84 


^s/i6 


9.40 


31-3 


62.5 


125.0 


156.3 


187.5 


218.8 


250.0 


281.3 


3.58 


I 


10.0 


33-3 


66.7 


100. 


133.3 


166.7 


200.0 


233-3 


266.7 


300.0 


3.36 



II INCHES Wide. 



inches. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


^ 


2-75 


9.17 


18.3 


27.5 


36.7 


45.8 


§5-2 


s/16 


3-44 


11. J 

13-8 


22.9 


34.4 


45.8 


68.' i 


68.8 


H 


4. "3 


27-5 


41-3 


55.0 


82.5 


7/16 


4.81 


16.0 


32.1 


48.1 


64.2 


i6.2 


96.3 


}4 


5.50 


18.3 


36.7 


55.0 


73.3 


91.7 


1 10.0 


9/16 


6.19 


20.6 


^^•i 


61.9 


82.5 


103,1 


123.8 


^, 


6.88 


22.9 


45-8 


68.8 


91.8 


114.6 


"37-5 


r 


7.56 


25.2 


50.4 


75.6 


100.8 


126.0 


i5>.3 


8.25 


27-5 


55.0 


82.5 


IIO.O 


137.5 


165.0 


r 


8.94 


29.8 


59-6 


89.4 


II9.2 


149.0 


178.8 


9.63 


32.1 


64.2 


96.3 


128.3 


160.4 


192.5 


*5/i6 


10.4 


34-4 


68.8 


103. 1 


137.5 


171.9 


206.3 


I 


11. 


36.7 


73-3 


1 10.0 


146.7 


183-3 


220.0 



lbs. lbs. 

64-2 73.3 

80.2 91.7 

96.3 I IIO.O 

1 12. 3 128.3 

128.3 146-7 

144.4 165.0 

160.4 1 183.3 

176.5 201.7 
192.5 220.0 

208.5 238.3 

224-6 256.7 

240.6 I 275.0 

256.7 293.3 



lbs. 


feet. 


82.5 


12.2 


103. 1 

123.8 


9.77 
8.15 


144.4 
165.0 


6.98 

6.II 


185.6 
206.3 
226.9 


5.43 
4-89 
4-44 


247-5 


4.07 


268.1 
288.8 


3.76 
3.49 


3094 
330.0 


3.26 
3.06 


1 



SQUARE IRON. 



239 



Table No. 75.— WEIGHT OP SQUARE IRON. 













Lkngth in Fbbt. 






w « 


SiDB. 


Sbct. 
Akba. 


















Lens;th 
to weigh 
























I 


a 


3 


4 


5 


6 


7 


8 


9 


X cwt. 


laches. 


aq. b. 


lbs. 


lbs. 


lbs. 


lU. 


lbs. 


lbs. 


lbs. 


lU. 


lbs. 


feet. 


X 


.0156 


.052 


.104 


.156 


.208 


.260 


.313 


'l^l 


.417 


.469 


2154 


3/16 


•035' 


.117 


.234 


•351 


.468 


.584 


.701 


.818 


.935 


1.05 


960.0 


}( 


.0625 


.208 


.4>7 


.625 


.833 


1.04 


1.25 


1.46 


1.67 


1.80 


537.6 


i/l6 


.0977 


.326 


•^^l 


.977 


1.30 


1.68 


'•95 


2.28 


2.60 


2.93 


343.8 


H 


.141 


.469 


.938 


I.4I 


1.58 


2.34 


2.81 


3.28 


3.75 


4.22 


238.3 


7/,6 


! .191 


.638 


1.28 


I.9I 


2.55 


3-19 


3.83 


4.46 


5.10 


5.74 


176.0 


H 


•25 


■833 


1.67 


2.50 


3.33 


4.17 


5.00 


5.83 


6.67 


7.50 


1344 


9/l« 


.316 


1.06 


2. II 


3- 16 


4.22 


5.27 


6.33 


7.38 


8.44 


9.49 


106.3 


H 


•391 


1.30 


2.60 


3.91 


5.21 


6.51 


7.81 


9.II 


10.4 


11.7 


85.9 


r 


, -473 


1.58 


3.J5 


473 


6.30 


7.88 


9-45 


II. 


12.6 


14.2 


71.0 


1.563 


1.88 


3-75 


5.63 


7.50 
8.80 


9.38 


"•3 


13." 


15.0 


16.9 


59.7 


T 


.661 


2.20 


4,40 


6.61 


II. 


13-2 


'54 


16.6 


19.8 


50.8 


.766 


2.55 


5.10 


7.66 


10.2 


12.8 


15.3 


17.9 


20.4 


23.0 


43.9 


*5/i6 


,879 


2.93 


5.86 


8.79 


II.7 


14.7 


17.6 


20.5 


234 


26.4 


38.2 


1 


I.CX> 


3.33 


6.67 


10.0 


13.3 


16.7 


20.0 


233 


26.7 


30.0 


33.6 


I »/i6 


. 1.13 


3.76 


7.53 


"•3 


15.I 


18.8 


22.6 


26.3 


30.1 


33.9 


29.7 


IH 


1.27 


4.22 


8.44 


12.7 


16.9 


21. 1 


^5-3 


29.5 


33.8 


38.0 


26.5 


I 3/16 


1 I.4I 


4.70 


9.40 


14. 1 


18.8 


23.5 


28.2 


32.9 


37.6 


42.3 


23.8 


iH 


: 1-56 


5.21 


10.4 


15.6 


20.8 


26.0 


31.3 


36.5 


41.7 


46.9 


21.5 


I 5/16 


' 1.72 


5.74 


1 1.5 


17.2 


23.0 


28.7 


344 


40.2 


45-9 


51.7 


19.5 


iH 


1 1'^ 


6.30 


12.6 


1S.9 


25.2 


31.5 


37.8 


44.1 


50.4 


56.7 


17.8 


I 7/x6 


' 2.07 


6.89 


13.8 


20.7 


27.6 


345 


4>-3 


48.2 


55.1 


62.0 


16.2 


1% 


1 2.25 


7.50 


15.0 


22.5 


30.0 


37.5 


45-0 


52.5 


60.0 


67.5 


14.9 


I 9/x6 


, 2.44 


8.14 


16.3 


24.4 


32.6 


40.7 


48.8 


57.0 


65.1 


73-2 


13-8 


IK 


< 2.64 


8.80 


17.6 


26.4 


35-^ 


44.0 


52.8 


61.6 


70.4 


79.2 


12.7 


I"/i6 


2.88 


9.60 


19.2 


28.8 


38.4 


48.0 


57.6 


67.2 


76.8 


86.4 


11.7 


I^ 


3.06 


10.2 


20.4 


30.6 


40.8 


51.0 


61.3 


71.4 


81.6 91.9 1 


II.O 


I '3/16 


' 3-29 


II. 


21.9 


32.9 


43.8 


54.8 


65.7 


76.7 


87.6 


98.6 


10.2 


I^ 


3.52 


11.7 


234 


35.2 


46.9 


58.6 


70.3 


82.0 


93.8 


105.5 


§■56 


H5/16 


: 3.75 


12.5 


25.0 


37.5 


50.1 


62.6 


25- » 


87.6 


100. 1 


112.6 


8.95 


2 


. 4.00 


'3-3 


26.7 


40.0 


53.3 


66.7 


80.0 


93.3 


106.7 


120.0 


8.40 


^>^ 


4.52 


15.1 


30.1 


45.2 


60.2 


75-3 


90.3 


105.4 


120.0 


135.5 


7.43 


25<^ 


5.06 


16.9 


33.8 


50.6 


67.1 


844 


101.3 


1 18. 1 


135.0 


151.9 


6.64 


2>i 


5.64 


18.8 


37.6 


56.4 


75.2 


94.0 


112.8 


131-6 


150.4 


169.2 


5.96 


2>^ 


6.25 


20.8 


41.7 


62.5 


833 


10.4 


125.0 


145.8 


166.6 


187.5 


5.38 


*>i 


,6.89 


23.0 


45-9 


68.9 


91.9 


114.9 


137.8 


160.8 


183.9 


206.7 


4-99 


2^ 


• 7.56 


25.2 


50.4 


75.6 


100.8 


126. 1 


151.3 


176.5 


201.7 


226.9 


444 


2^ 


S.27 


27.6 


55.1 


82.7 


1 10.2 


137.8 


165.3 


192.9 


220.4 


248.0 


4.06 


3 


9,00 


30.0 


60.0 


90.0 


120.0 


150.0 


180.0 


210.0 


240.0 


270.0 


3.73 


3X 


} 10.6 


35-2 


70.4 


105.6 


140.8 


176.0 


211. 3 


246.5 


281.7 


316.9 


3.17 


3>^ 


1 12.3 


40.8 


81.7 


122.5 


163.3 


204.2 


245.0 


285.8 


326.7 


367.5 


2.73 


3^ 


14. 1 


46.9 


93.8 


140.6 


187.5 


234.4 


281.3 


328.1 


375.0 421.9 


2.38 


4 


1 16.0 


53.3 


106.7 


160.0 


213-3 


266.7 


320.0 


3730 


426.0 1 480.0 


2.10 


4^ 


. 18. 1 


60.2 


120.4 


180.6 


240.8 


301. 1 


361.2 


421.5 


481.7 541.9 


1.86 


4^ 


20.3 


67.5 


1350 


202.5 


270.0 


337.5 


405.0 


472.5 


540.0 


607.5 


1.66 


4¥ 


1 22.6 


Z5-2 


150.4 


225.6 


300.8 


376.1 


451.3 


526.5 


601.7 


676.9 


1.49 


5 


25.0 


83.3 


166.7 


250.0 


333-3 


416.7 


500.0 


583.3 


666.7 


750.0 


».34 


^K 


27.6 


91.9 


183.8 


275.6 


367.5 


459.4 


551.3 


643.1 


735-0 


826.9 


1. 21 


5^ 


30.3 


100.8 


201.7 


302.5 


403.3 


504.2 


605.0 


705.8 


806.7 


907.5 


I.U 


53^ 


33.1 


1 10.2 


220.4 


330.6 


440.8 


551.0 


661.3 


771.5 


881.7 


991.8 


1.02 


6 


36.0 


120.0 


240.0 


360.0 


480.0 


600.0 


720.0 


840.0 


960.0 


1080 


.933 



240 



WEIGHT OF METALS. 



Table No. 76.— WEIGHT OF ROUND IRON. 













Lbngth in 


Feet. 








T -_!_ 


DiAM. 


Sect. 
Area. 




















Length 




















to weig}i 






z 


a 


3 


4 


5 


6 


7 


8 


9 


I cwt. 


inches. 

1 


1 sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


feet. 


X 


1 .0123 


.041 


.082 


.123 


.164 
.368 


.205 


.245 


.286 


.327 


.368 


2738 


3/16 


.0276 


.092 


.184 


.276 


.460 


•552 


•644 


.736 


.828 


1217 


H 


.0491 


.164 


.327 


.491 


.655 


.818 


.982 


115 


I.3I 


1.47 


684.4 


5/16 


.0767 


.256 


.511 


.767 


1.02 


1.28 


1.53 


1.79 


2.04 


2.30 


438.1 


H 


.110 


.368 


.736 


1. 10 


1-47 


1.84 


2.21 


2.58 


2.94 


3.3> 


305.4 


7/1 s 


.150 


.501 


I.OO 


1.50 


2.00 


2.51 


3.01 


3-5i 


4.01 


4.51 


224.0 


>i 


.196 


.654 


1. 21 


1.96 


2.62 


327 


3.93 


4.58 


5-23 


5.89 


171.4 


9/16 


.24S 


.828 


1.66 


2.49 


3.31 


4.14 


4.97 


5.80 


6.63 


7.46 


135.5 


^. 


.307 


1.02 


2.05 


3-07 


4.09 


5.II 


6.14 


7.16 
8.66 


8.18 


9.20 


109.5 


r 


.371 


1.24 


2.48 


3.71 


4.9s 


6.19 


7.42 


9.90 


II. I 


90.6 


•442 


1.47 


2.94 


4.42 


5.89 


7.36 : 


8.83 


10.3 


11.8 


13.3 


76.0 


r 


.518 


1-73 


346 


519 


6.91 


8.64 


10.4 


12. 1 


13.8 


15.6 


70.5 


.601 


2.00 


4.01 


6.01 


8.02 


10.0 


12.0 


14.0 


16.0 


18.0 


55-9 


'5/16 


.690 


2.30 


4.60 


6.90 


9.20 


11.5 1 


13.8 


16. 1 


18.4 


20.7 


48.7 


I 


.785 


2.62' 


' 5.24 


7.85 


10.5 


13." 1 


15.7 


18.3 


20.9 


23.6 


• 42.8 


I ^/i6 


.887 


2.96 


5.91 


8.87 


11.8 


14.8 , 


17.7 


20.7 


23.6 


26.6 


37.9 


iH 


•994 


3-31 


6.63 


9-94 


^3-3 


16.6 


19.9 


23.2 


26.5 


29.8 


33-8 


I 3/16 


I. II 


3.69 


7.38 


II. I 


14.8 


18.5 


22.2 


25.8 


29.5 


33-2 


30.3 


'^, 


1-23 


4.09 


8.18 


12.3 


16.4 


20.5 ' 


24.5 


28.6 


32.7 


36.8 


27.3 


I 5/x6 


^•35 


45' 


9.02 


13.5 


18.0 


22.6 


27.1 


31.6 


36.1. 


40.6 


24.9 


I^ 


1.48 


4.95 


9.90 


14.9 


19.8 


24.8 


29.7 


34.6 


39.6 


46.6 


22.7 


I 7/x6 


1.62 


5.08 


10.2 


16.2 


20.3 


25.9 


32.5 


35. 5 


40.6 


48.7 


20.7 


I>i 


1.77 


5.89 


II. 8 


17.7 


23.6 


29.5 


35.3 


41.2 


47-1 


53-0 


19.0 


I 9/16 


1.92 


6.39 


12.8 


19.2 


25.6 


32.0 \ 


38.4 


44.7 


51.1 


57.5 


'7.5 


ifi 


2.07 


6.91 


13.8 


20.7 


27.7 


34.6, 


41.5 


48.4 


55-3 


62.9 


16.2 


I"A6 


2.24 


7.46 


14.9 


22.4 


29.8 


37.31 


^\ 


52.2 


59.6 


67.1 


15.0 


I^ 


2.41 


8.02 


16.0 


24.1 


32.1 


40.1 1 


56.1 


64.1 


72.2 


13.9 


I » 3/16 


2.58 


8.60 


17.2 


25.8 


34-4 


43.0 


51.6 


60.2 


68.8 


77-4 


13.0 


I^ 


2.76 


9.20 


18.4 


27.6 


36.8 


46.0 


55.2 


64.4 


73.6 


82.8 


12.2 


1*5/16 


2.95 


9.83 


19.7 


29.5 


39.3 


49.1 


59.0 


68.8 


l^'^o 


88.4 


11.4 


2 


3.14 


10.5 


20.9 


31.4 


41.9 


52.4 j 


62.8 


73-3 


83.8 


94.3 


10.7 


2>^ 


3.55 


11.8 


23.6 


35-5 


47.3 


59.1 ; 


70.9 


82.8 


94.6 


106.4 


9.47 


2X 


3.98 


133 


26.5 


39.8 


53.0 


66.3 


Z2-5 


92.8 


106.0 


"93 


8.44 


2^ 


4.43 


14.8 


29.5 


44.3 


59.1 


73.8, 


88.6 


103.3 


1 18. 1 


132.9 


7-59 


2;^ 


4.91 


16.4 


32.7 


49.1 


65.5 


81.8, 


98.2 


1 14. 5 


130.9 


147.3 


6.84 


2>i 


5.41 


18.0 


36.1 


54.1 


72.2 


90.2 


108.2 


126.2 


144.3 


162.3 


6.21 


2.V 


5-94 


19.8 


39.6 


59.4 


79.2 


99.0 


118.8 


138.5 


158.4 


178.2 


5.66 


2J^ 


6.49 


21.6 


43-3 


64.9 


86.6 


108.2 


129.8 


151.5 


1 73. 1 


194.8 


5.18 


3 


7.07 


23.6 


47.1 


70.7 


94.3 


117.8 


141.4 


164.9 


188.5 


212.1 


4.7s 


3^ 


8.30 


27.7 


55-3 


83.0 


no. 4 


138.3 ' 


165.9 


193.6 


221.2 


248.9 


4.05 


3>^ 


9.62 


32.1 


64.1 


96.2 


128.3 


160.4 


192.4 


224.5 


256.6 


288.6 


3-49 


3^ 


II.O 


33-5 


23-^ 


1 10.4 


147.3 


164. 1 


220.9 


257.7 


294.5 


33'.3 


3.04 


4 


12.6 


41.9 


83.8 


125.7 


167.6 


209.4 


251.3 


293.2 


335.0 


377.0 


2.67 


^^ 


14.2 


47.3 


94.6 


141. 9 


189. 1 


236.4 


283.7 


331.0 


378.3 


425.6 


2.37 


4H 


15.9 


53.0 


106.0 


159.0 


212.1 


265.1 


319. 1 


371. 1 


424.1 


477.1 


2.11 


aH 


17.7 


59.1 


1 18. 1 


177.2 


236.3 


295.3 


354.4 


413.5 


472.5 


531.6 


1.90 


5 


19.6 


65.5 


130.9 


196.4 


261.8 


327.3 


392.7 


458.2 


523.6 


589.1 


1.71 


5^ 


21.7 


72.2 


144.3 


216.5 


288.6 


360.8 


432.9 


505.1 


577-3 


649.4 


1.55 


5>^ 


23.8 


79.2 


158.4 


237.6 


316.7 


396.0 


475.2 


554.3 


633.6 


712.7 


I.41 


\^ 


26.0 


86.6 


173. 1 


259.7 


346.2 


432.8 


519.3 


605.9 


692.4 


779.0 


1.29 


6 


28.3 


94.2 


188.5 


282.7 


377.0 


471.2 


565.5 


659.7 


754.0 


848.2 


1. 19 



ROUND IRON. 



241 



Weight of Round Iron. 







Length in Feet. 






T\m a «.* 


Sect. 




L«n^h 
to weigh 


^"^*'- 1 Arka. I 
























z 


a 


3 


4 


s 


6 


7 


8 


9 


I ton. 


'inches. 


sq. in. 

33-2 


cwts. 
.9876 


cwts. 

1.975 


cwts. 

2.963 


cwts. 

3.950 


cwts. 

4.938 


cwts. 

5.926 


cwts. 

6.613 


cwts. 

7.901 


cwts. 

8.888 


feet 
20.2 


6>4 


7 


38.5 


1. 145 


2.291 


3.436 


4.582 


5727 


6.872 


8.018 


9.163 


10.31 


17.5 


7}i 


44.2 


I.315 


2.629 


3-944 


5.258 


6.573 


7.887 


9.202 


10.52 


11.84 


15.2 


8 


503 


1.496 


2.992 


4.448 


5.984 


7.480 


8.976 


10.47 


11.97 


13.46 


13.4 


8^ 


56.7 


1.689 


3.378 


5.067 


6.756 


8.444 


10.13 


11.82 


13.50 


15.20 


II.8 


9 


63.6 


1.893 


3.786 


5.680 


7572 


9.46 


11.36 


13-25 


'5i4 


17.04 


10.6 


9^ 


70.9 


2. 1 10 


4.220 


6.329 


8.440 


10.55 


12.66 


14.77 


16.88 


18.99 


9.48 


10 


78.5 


2.338 


4.676 


7.012 


9.352 


11.69 


1403 


16.37 


18.70 


21.04 


8.56 


10)^ 


86.6 


2.577 


4754 


Z-7?' 


10.31 


12.89 


15.46 


18.04 


19.02 


23-19 


7-76 


II 


1 950 


2.828 


5.656 


8.485 


II. 31 


14.14 


16.97 


19.80 


22.62 


25.46 


7.07 


Il>^ 


; 103- 9 


3.088 


6.176 


9.265 


12.35 


15-44 


18.53 


21.62 


24.70 


27.80 


6.47 


1 12 


! 1131 


3366 


6.732 


10.10 


13.46 


16.83 


20.20 


23-56 


26.93 


30.29 


5.94 


I2>^ 


; 122.7 


3656 


7.312 


10.96 


14.62 


18.28 


21.91 


25-59 


29.25 


32.90 


5.48 


13 


'132.7 


3.950 


7.900 


tiM 


15.80 


19.75 


23.70 


27.65 


31.60 


35.15 
38.34 


5.06 


i3>^ 


143. 1 


4.260 


8.520 


12.78 


17.04 


21.30 


25.56 


29.82 


34.08 


4.70 


14 


153.9 


4.581 


9.162 


13.74 


18.32 


22.90 


26.49 


32.07 


36.65 


41.23 


437 


14^ 


1 165.1 


4.915 


9.830 


1474 


19.66 


2458 


28.49 


34-41 


39.32 


44.24 


4.07 


i'5 


176.7 


5.259 


10.52 


15.78 


21.04 


26.30 


31-46 


36.81 


42.08 


47.33 


3.80 


15;^ 


1 188.7 


5.616 


11.23 


16.85 


22.46 


28.08 


32.70 


39-31 
41.89 


44.92 


50.54 


3.56 


16 


201. 1 


5.984 


11.97 


17.95 


23.93 


29.92 


35.90 


47.88 


53.86 


3.34 


16X 


213.8 


6.364 


12.73 


19.09 


25.46 


31.82 


38.18 


44.55 


50.92 


57.28 


3-14 


17 


227.0 


6.755 


13.51 


20.27 


27.02 


33-78 


40.53 


47-29 


54.04 


60.80 


2.96 


;I7^ 


1 240.5 


7.»59 


14.32 


21.48 


28.64 


35-5? 


42.95 


50.11 


57.28 


6443 


2.79 


iiS 


254.5 
2S3.5 


Z-573 


'5i5 


22.72 


30.29 


37-86 


45-44 


5301 


60.60 


68.16 


2.64 


19 


8.438 


16.88 


25.32 


33.75 


42.19 


50.63 


59.03 


67.52 


75.94 


2.37 


ao 


314-2 


9350 


18.70 


28.05 


37.40 


46.75 


56.10 


65.45 


7480 


84.15 


2.14 


21 


346.4 


10.31 


20.62 


30.93 


41.23 


51.54 


^J'li 


72.16 


82.47 


92.78 


1-94 


22 I 


380.1 


II. 31 


22.63 


33-94 


45.25 


56.57 


67.88 


79.19 
86.56 


90.51 


101.8 


1.77 


23 1 4«5-5 


12.37 


2473 


37.10 


49.46 


61.83 


Z^'g 


93.92 


III. 3 


1.62 


24 , 452.4 

1 


13-46 


26.93 


40.39 


53.86 


67.32 1 


80.78 


94.25 


107.7 


121.3 


1.49 



16 



242 



WEIGHT OF METALS. 



Table No. 77.— -WEIGHT OP ANGLE-IRON AND TEE-IRON. 

I Foot in Length. 

Note. — When the base or the web tapers in section, the mean thickness is to be measured. 



Thick- 

NSSS. 



Sum of the Width and Depth ik Inches. 



inches. 
3/16 

5/16 



I^ 



lbs. 

.81 
1.04 
1.24 



iH 



lbs. 

.62 

.89 
I.15 

1.37 



^H 



lbs. 

.68 

.97 
1.25 

1.50 



^H 



lbs. 

•73 
1.05 

1.36 

1.63 



lbs. 

.78 

I.I3 

1.46 
1.76 



^>i 



lbs. 

.83 
1. 21 

1.56 

1.89 



2H 


^H 


lbs. 


lbs. 


.88 


.94 


1.29 


1.37 


1.67 


1.77 


2.02 


2.15 



^'A 



lbs. 

.99 
1.45 

1.88 

2.28 



^H 



lbs. 

1.04 
1.52 
1.98 
2.41 



2U 



lbs. 

1.09 
1.60 
2.08 
2.54 



3/i6 
S/16 
7/16 



2% 



1. 14 

1.68 
2.19 
2.67 

3.13 
3.57 



1.20 
1.76 
2.29 
2.80 
3.28 

3.75 



3'A 



aX 



sH 



zA 



1.25 
1.84 

2.40 
2.93 

3-44 
3.93 



1.30 
1.91 

2.50 

306 

3-59 
4. 1 1 



1.45 
1.99 

2.60 
3.19 

3.75 
4.29 



1. 41 

2.07 

2.71 

3.32 

3-91 
4.48 



ZH 



1.46 

2. IS 
2.81 

3-45 
4.06 

4.66 



zH 



1.51 
2.23 
2.92 

3.58 

4-22 
4.84 



3% 



1.56 
2.30 
3.02 

371 
4.38 
5.02 



1.62 
2.38 

3.13 

3.84 

4.53 
5.20 



A% 



1.72 
2.54 

3-33 
4.10 

4.84 

5.56 



3/16 

S/x6 

H 

7/16 

91x6 



4H 



aU 



2.70 

3.S4 
436 
5.16 
5.92 
6.67 

7.38 



2.85 

3-75 
4.62 

5-47 
6.29 

7.08 

7.85 



3.01 

4.88 

5.78 
6.65 
7.50 
8.32 



5X 



3.16 
4.17 

5- 14 
6.09 

7.02 

.92 

.79 



I 



sH 



5H 



6X 



6>i 



3.32 
4.38 
5.40 
6.41 

7.38 

8.33 
9.26 



3.48 

4.58 
5.66 
6.72 

7.75 
8.75 
9.73 



3.63 
4-79 
5.92 

7.03 
8.11 

9.17 

10.20 



3 79 
5.00 

6.18 

7-34 
8.48 

9.58 
10.66 



3.95 
5.21 

6.45 

7.66 

8.84 

10.00 

II. 13 



6^ 



4.10 
5.42 
6.71 

7.97 
9.21 

10.42 

11.60 



4.26 

5-63 
6.97 

8.28 

10.83 
12.07 



5/16 

H 

7/16 

9/x6 
H 



1% 



5-83 
7.23 
8.59 

9.93 
11.25 

12.54 

i^8o 



rA 


7^ 


6.04 


6.25 


7-49 


7.75 


8.91 


9.22 


10.30 


10.66 


11.67 


12.08 


1301 


13.48 


14.32 


14.84 



8 



8X 



8K 



6.46 
8.01 

9.53 
11.03 

12.50 

13.94 
15.36 



6.67 

8.27 

9.84 

11.39 
12.92 

14.41 

15.89 



6.88 

8.53 
10.16 

11.76 

'3-33 
15.88 

16.41 



8^ 



9X 



^% 



7.08 

8.79 
10.47 

12.12 

13.75 
15.35 
16.93 



7.39 
9.05 
10.78 
12.49 
14.17 
15.82 

17.45 



750 

9.31 
11.09 

12.85 

14.58 

16.29 

17.97 



7.71 

9.57 
II. 41 

13.22 

15.00 

16.76 

18.49 



9^ 



7.92 

9.83 
11.72 

13.58 
15.42 
17.23 
19.01 



7/16 

'A 

91x6 



zo 


loK 


12.03 


12.66 


13.95 


14.67 


15.83 


16.67 


17.70 


18.63 


19.53 


20.57 


23.13 


24.38 



ZI 



">i 



12 



13.28 
15.40 
17.50 

19.57 
21.61 

25.63 



13.91 
16.13 

18.33 
20.51 

22.66 

26.88 



14.53 
16.86 

19.17 

21.44 

23.70 

28.13 



ia>^ 



17.59 
20.00 

22.38 

24.74 

29.37 



13 13K 



14 



14^ 



18.31 
20.84 

23.31 
25.78 

30.63 



19.04 
21.67 
24.25 
26.83 
31.88 



19.77 
22.50 
25.19 
27.87 

33.13 



20.50 

2334 
26.12 

28.91 
34.38 



15 



21.22 
24.17 
27.06 
29.95 
3563 






12 



25.70 
28.13 

32.45 
36.67 



izV 



24.74 

29.37 
33.91 
38.33 



13 



13K 



25.78 
30.63 

35.36 
40.00 



26.83 
31.88 
36.82 
41.67 



14 



27.87 

38.28 

43-33 



15 



29.95 
3563 
41.19 

46.67 



16 



17 



18 



19 



20 



32.03 

38.13 
44.12 

50.00 



34.12 
40.63 
47.02 

53-33 



36.20 

41.13 

49.95 
56.67 



38.28 

43.63 
52.87 
60.00 



40.36 
46.13 

55.78 

63.33 



WROUGHT-IRON PLATES. 



243 



Table No. 78.— WEIGHT OF WROUGHT-IRON PLATES. 



1 


Sect. 
Akra, 
when 
I foot 






Area in Square Feet. 


Number 


Thick- 
mss. 








of sq. ft. 
in 






















wide. 


I 


2 


3 


4 


5 


6 


7 


8 


9 


X ton. 


inch«. 


sq. in. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


sq. feet. 


X 


3.00 


10.0 


20.0 


30.0 


40.0 


50.0 


60.0 


70.0 


80.0 


90.0 


224.0 


5/16 


3.75 


12.5 


25.0 


37.5 


50.0 


62.5 


75.0 


87.5 


100. 


112.5 


179.2 


H 


4.50 


15.0 


30.0 


45.0 


60.0 


75.0 


90.0 


105.0 


120.0 


I35-0 


149.3 


7/16 


5.20 


17.5 


35.0 


52.5 


70.0 


87.5 


105.0 


122.5 


140.0 


157.5 
180.0 


128.0 


^ 


6.00 


20.0 


40.0 


60.0 


80.0 


100.0 


120.0 


140.0 


160.0 


112.0 


9/16 


I 6.75 


22.5 


45.0 


67. 5 


90.0 


112.5 , 


135.0 


150.0 


180.0 


202.5 


99.67 


H 


l^"" 


25.0 


50.0 


75.0 


lOO.O 


125.0 1 


150.0 


175.0 


200.0 


225.0 


89.60 


«»A6 


1 8.25 


27.5 


55.0 


82.5 


ZIO.O 


137.5 


165.0 


192.5 


220.0 


247.5 


81.45 


H 


9.00 


30.0 


60.0 


90.0 


120.0 


150.0 


180.0 


210.0 


240.0 


270.0 


74.67 


»3/i6 


9.75 


32.5 


65.0 


97.5 


130.0 


162.5 


195.0 


227.5 


260.0 


292.5 


68.92 


^ 


11.50 


35.0 


70.0 


105.0 


140.0 


175.0 


210.0 


245.0 


280.0 


315.0 


64.00 


^s/x6 


11.25 


37.5 


75.0 


112.5 


150.0 


187.5 ' 


225.0 


262.5 


300.0 


337.5 


59-73 


X- 


12.00 


40.0 


80.0 


120.0 


160.0 


200.0 


240.0 


280.0 


320.0 


360.0 


56.00 


I V«6 


12.75 


42.5 


85.0 


127.5 


170.0 


212.5 


255.0 


297.5 


3+0.0 


382.5 


52.71 


I^ 


13.50 


450 


90.0 


1350 


z8o.o 


225.0 


270.0 


3150 


360.0 


405.0 


49.78 


I 3/z6 


14,25 


47.5 


95.0 


142.5 


190.0 


237.5 


285.0 


332.5 


380.0 


427.5 


47.16 


x>< 


15.0 


50.0 


lOO.O 


150.0 


200.0 


250.0 


300.0 


350.0 


400.0 


450.0 


44.80 


I^ 


16.5 


55-0 


1 10.0 


165.0 


220.0 


275.0 1 


330.0 


385.0 


440.0 


4950 


40.73 


I^ 


1 18.0 


60.0 


120.0 


iSo.o 


240.0 


300.0 


360.0 


420.0 


480.0 


540.0 


37.33 


l|F 


21.0 

1 


70.0 


140.0 


210.0 


280.0 


350.0 


420.0 


490.0 


560.0 


630.0 


32.00 


2 


24.0 


80.0 


160.0 


240.0 


320.0 


400.0 


480.0 


560.0 


640.0 


720.0 


28.00 






cwts. 


cwls. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 




2}4 


30 


.893 


1.79 


2.68 


3-57 


4.46, 


§■36 


6.25 


Z-'^ 


8.04 


25.40 


3 


36 


1.07 


2.14 


3.21 


4.29 


5.36 


6.64 


7.50 


8.57 


9.64 


18.67 


3->^ 


^ 


1.25 


2.50 


3.75 


5.00 


6.25 


^50 
s.57 


8-75 


10.00 


11.25 


16.00 


4 


48 


1.43 


2.86 


4.29 


571 


7.14 


10.00 


"•53 


12.86 


14.00 


4>i 


54 


1. 61 


3.21 


4.82 


6.43 


8.04 


9.64 


11.25 


12.86 


14.46 


12.44 


5 


60 


1.79 


3.57 


5.36 


7.14 


8.93 


10.71 


12.50 


14.29 


16.07 


11.20 


5>^ 


66 


1.96 


3.93 


5.S9 


7.86 


9.82 


11.79 


13.75 


15.71 


17.68 


10.18 


6 


72 


2.14 


4.29 


6.43 


8.57 


10.71 


12.86 


15.00 


17.14 


19.29 


9-33 


7 


84 


2.50 


5.00 


Z'5° 


10.00 


12.50 


1500 


17.50 


20.00 


22.50 


8.00 


8 


95 


2.86 


5.71 


8.57 


"43 


10.29 


17.14 


20.00 


22.86 


25.71 


7.00 


9 


108 


3-21 


6.43 


9.64 


12.86 


16.07 


19.29 


22.50 


25.71 


28.93 


6.22 


10 


120 


3.57 


7.14 


10.71 


14.29 


12.86 


21.43 


25.00 


28.56 


32.14 


5.60 


Ji 


132 


3-93 


7.86 


11.79 


15.71 


19.64 


2357 


27.50 


31.43 


35.36 


5.09 


12 


144 


4.29 


8.57 


Z2.86 


17.14 
18.57 


21.43 


25.71 


30.00 


34.29 


38.57 


4.67 


J3 


'5^ 


4.64 


9.29 


13.93 


23.21 


27.86 


32.50 


37.14 


41.79 


431 


14 


168 


5.00 


10.00 


15.00 


20.00 


25.00 1 


30.00 


3500 


40.00 


45.00 
48.21 


4.00 


•' il 


180 


5.36 


10.71 


16.07 


21.43 


26.79 

1 


32.14 


37.50 


42.86 


3-73 



244 



WEIGHT OF METAI^S. 



Table No. 79.— WEIGHT OF SHEET IRON. 

AT 480 LBS. PER CUBIC FOOT. 

According to Wire-gauge used in South Staffordshire (Table No. 17). 



Thickness. 



Area in Square Fbrt. 



B.W.G. 
32 

30 
29 

28 
27 
26 

25 

24 

23 
22 

21 

20 

19 
18 

17 

16 

IS 
14 
13 

12 
II 

10 

9 
8 

7 
6 

5 

4 

3 

2 

I 



inch. 

.0125 
.0141 
.0156 
.0172 

.0188 
.0203 
.0219 
.0234 

.0250 
.0281 

.0313 
•0344 

.0375 
.0438 

.0500 
•0563 

.0625 
.0750 
.0875 
.1000 

.1125 
.1250 
.1406 

.1563 

.1719 

.1875 

.2031 

.2188 
.2344 

.2^00 
.2813 
.3125 



lbs. 

.500 
.562 
.625 

.688 

.750 
.813 
.875 

.938 

1. 00 

113 

I.2q 
1.38 

I. SO 

I.7S 
2.00 

2.25 

2.50 
3-00 

3- SO 
4.00 

4.50 
5.00 

5.63 
6.25 

6.88 
7.50 

813 

8.75 

9.38 
10.0 

11.25 

12.5 



lbs. 

1. 00 

I.I3 
1.25 

1.38 

1.50 
1.63 

1.88 

2.00 
2.25 
2.50 

2.75 

300 

3. SO 
4.00 

4.50 

5.00 
6.00 
7.00 
8.00 

9.00 

lO.O 

"3 
12.5 

13.8 
15.0 
16.3 

17.S 

18.8 
20.0 
22.5 
25.0 



lbs. 

1.50 
1.69 
1.88 
2.06 

2.25 
2.44 
2.63 
2.81 

3.00 
3.38 
3.7s 
4.13 

4.50 

5.25 
6.00 

6.75 

7. SO 
9.00 
10.5 
12.0 

13.5. 
15.0 

16.9 

16.8 

20.6 
22.5 

24.4 
26.3 

28.1 
30.0 
33.8 
37.5 



lbs. 

2.00 
2.25 
2.50 
2.75 

3.00 

3.2s 
3.50 

3.7s 
4.00 

4.50 
5.00 

5.50 

6.00 
7.00 
8.00 
9.00 

10.0 
12.0 
14.0 
16.0 

18.0 
20.0 
22.5 
25.0 

27-5 
30.0 

32. S 

3S.O 

37. S 

40.0 
45.0 
50.0 



lbs. 

2.^0 
2.81 

313 
3-44 

3-75 
4.06 

4.38 
4.69 

S-oo 

S.63 
6.25 

6.88 

7.50 

8.75 
10. o 

"•3 

12.5 

150 

17.5 
20.0 

22.5 

2q.O 
25.1 

31-3 

34.4 

37.S 
40.6 

43.8 

46.9 
50.0 

56.3 
62.S 



lbs. 

3.00 
3.38 
3.75 
4.13 

4- JO 
4.88 

5.2s 
S.63 

6.00 
6.75 

7.50 
8.25 

9.00 
10.5 
12.0 

13.5 

IJ.O 

18.0 
21.0 
24.0 

27.0 
30.0 

33.8 

37. S 

41.3 
45.0 

48.8 
S2.S 

56.3 

60.0 

67.5 
75.0 



lbs. 

3-50 
3.94 
4.38 
4.81 

5-2S 

5.69 

6.13 

6.56 

7.00 

7.88 

8.7s 

9.63 

10.5 
12.3 

14.0 

15.8 
I7.S 

21.0 

24. S 
28.0 

31.5 
3S.O 

49-4 
43.8 

48.1 

S2.5 
56.9 

61.3 

65.6 
70.0 
78.8 

87. S 



8 



lbs. 

4.00 
4.50 
5.00 

S-So 

6.00 
6.50 
7.00 
7.50 

8.00 
9.00 

lO.O 
II.O 

12.0 
14.0 
16.0 
18.0 

20.0 
24.0 
28.0 
32.0 

36.0 
40.0 
45.0 
50.0 

SS.o 
60.0 
65.0 
70.0 

7S.0 

80.0 

90.0 

100.0 



lbs. 

4.50 
5.06 

5.63 
6.19 

6.75 

7.31 
7.88 

8.44 

9.00 
10. 1 

"3 
12.4 

13.S 

18.0 
20.3 

22.5 
27.0 

31.S 
36.0 

40.5 
45.0 

50.6 

S6.3 

61.9 
67.5 
72.1 
78.8 

84.4 
90.0 

101.3 

112.5 



Number 
of sq. ft. 
in I ton. 

sq. ft 

4480 
3986 
3584 

32S6 

2987 
275s 

2560 

2388 

2240 
1982 

1792 
1623 

1493 
1280 

1 120 

996 
896 

747 
640 

560 

498 
448 
398 
358 

326 

299 
276 

256 

239 

224 

199 
179 



IRON SHEETS. 



24s 



Table No. 80.— WEIGHT OF BLACK AND GALVANIZED 

IRON SHEETS. 



(Morton's Table, founded upon Sir Joseph Whitworth & Co.*s Standard 

Birmingham Wire-Gauge.) 

XoTE. — ^The numbers on Holtzapflfel's wire-gauge are applied to the thicknesses 

on Whitworth's gauge. 



Gauge of Black Sheets. 


Approximate number of 
sqtiare feet in i ton. 


Gauge of Black Sheets. 


Approximate ntmiber of 
square feet in x ton. 


Wire- 


Thickness. 


Black 


Galvanized 


Wire- 


Thickness. 


Black 


Galvanized 


Gauge. 


Sheets. 


Sheets. 


Gauge. 


Sheets. 


Sheets. 


No. 


inch. 


square feet 


square feet. 


No. 


inch. 


square feet 


square feet 


I 


.300 


187 


185 


17 


.060 


933 


876 


2 


.280 


200 


197 


18 


.050 


1120 


1038 


3 


.260 


215 


212 


19 


.040 


1400 


1274 


4 


.240 


233 


229 


20 


.036 


1556 


1403 


5 


.220 


^§^ 


250 


21 


.032 


1750 


1558 


6 


.200 


2^^ 


275 


22 


.028 


2000 


1753 


7 


.180 


3" 


304 


23 


.024 


2333 


2004 


8 


.165 


339 


331 


24 


.022 


2545 


2159 


9 


.150 


373 


363 


^§ 


.020 


2800 


2339 


10 


.135 


415 


403 


26 


.0x8 


3111 


?^ 


II 


.120 


467 


452 


H 


.016 


3500 


12 


.110 


509 


491 


28 


.014 


4000 


3122 


'3 


.095 


589 


566 


29 


.013 


4308 


3306 


14 


.085 


659 


630 


30 


.012 


4667 


3513 


15 


.070 


800 


757 


31 


.010 


5600 


4017 


16 

1 


.065 


862 


813 


32 


.009 


6222 


4327 



246 



WEIGHT OF METALS. 



Table No. 81.— WEIGHT OF HOOP IRON. 



I FOOT IN LENGTH. 



According to Wire-gauge used in South Staffordshire. 



Thicj 








Width in 


Inches. 








CNBSS. 


H 


^ 


H 


I 


^}i 


IX 


iH 


i>4 


B.W.G. 


inches. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 1 


lbs. 


21 

20 

19 


■0344 

.0375 
.0438 


.0716 
.0781 
.0911 


.0861 
.0938 

.109 


.100 
.109 
.128 


.115 

.146 


.129 
.141 
.164 


.144 
.156 
.182 


.158 
.172 
.200 


.172 

.188 

.219 


18 
16 


.0500 

.0563 
.0625 


.104 
.117 
.130 


.125 

.141 
.156 


.146 
.164 
.182 


.167 

.188 

.208 


.188 
.211 

.234 


.208 

.234 
.260 


.229 
.258 
.286 


.250 
.281 

.313 


15 

13 


.0750 
.0875 
.1000 


.156 
!208 


.188 

.219 
.250 


.219 
.256 
.292 


.250 

.293 
.333 


.281 

.329 
.375 


•313 
.366 
.416 


.344 
.402 

•458 


.375 

.438 

.500 


12 
II 
10 


.1125 
.1250 
.I40& 


.234 

.260 

.293 


.281 

.313 
.352 


.328 
•365 

.410 


.375 
.417 
.469 


.422 
.469 
.527 


.469 
.521 

.586 


.516 

•573 
.645 


.563 

.625 

.703 


7 


.1563 
.1719 

.1875 


.326 

.358 

.391 


.391 

.430 
.469 


.456 

.501 

.547 


.522 

.573 

.625 


•587 
.644 

•703 


.652 
.716 
.781 


.717 
.788 

.859 


•783 
.859 

.938 


6 

5 

4 


.2031 
.2188 

.234+ 


■423 
.488 


.508 

:r8^ 


.030 • 

.683 


.677 
.729 
.781 


.762 
.820 

.879 


.836 
.912 

.977 


.931 
10.0 

10.7 


1.02 
1.09 

I.I7 




KNBSS. 








Width in 


Inches. 








IHIC 


^H 


Il< 


I^ 


a 


2X 


2>i 


2J< 


3 


B.W.G. 


inches. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


21 

20 

19 


.0344 

.0375 
.0438 


.197 

.203 

.238 


.201 
.219 
.257 


.215 

.224 

.274 


.229 

.250 
.292 


.258 
.251 
.328 


.287 

.313 
.365 


•315 

•344 
.400 


.344 

•375 
.437 


18 

17 
16 


.0500 
.0563 
.0625 


.271 

.305 

•339 


.292 

.328 
.365 


.312 

.351 
.391 


•333 
•375 
.417 


.375 
.422 

.469 


.417 
.469 
.521 


.458 
.516 

.573 


.500 

.563 
.625 


15 

13 


.0750 
.0875 
.1000 


.307 
.475 
•543 


.438 
.584 


.469 

•549 
.626 


.585 
.667 


.562 
.658 
.750 


.625 

.833 


.687 
.804 
.917 


.750 
.875 

I.OO 


12 
II 
10 


.1125 
.1250 
.1406 


.609 
.677 
.762 


.656 
.729 
.820 


.703 
.781 

.879 


.750 

•833 
.938 


.842 

.937 
1.06 


.938 

1.04 

1.17 


1.03 
1. 15 
1.29 


I.I3 

1.16 


7 


.1563 
.1719 
.1875 


.848 

.931 
1.02 


•913 

1. 00 
1.09 


.978 
1.07 

1.17 


1.04 
i.iS 
1.25 


I.I7 
1.29 
1.41 


1.30 

1^43 
1.56 


1.58 
1.72 


1.56 

1.72 

1.88 


6 

5 
4 


.2031 
.2188 

.2344 


1. 10 
1.19 
1.27 


1.28 

1.37 


1.27 

1.37 
1.46 


1-35 
1.46 

1.56 


1.64 
1.76 


1.69 
1.82 

i'9S 


1.86 
2.C0 
2.15 


2.03 
2.19 

2.35 



WARRINGTON IRON WIRE. 



247 



Table No. 82.— WEIGHT AND STRENGTH OF WARRINGTON 

IRON WIRE. 



Table of Wire manufactured by Rylands Brothers. 
Note.— The Wire-Gauge is that of Rylands Brothers. 



1 

i 


1 
















Specific 


1 

Size on ; 






Weight of II 


Length of j 


Breaking Strain. | 


Density, 


Wire- 1 
Gauge. 


Diameter. 


100 Yds. 


iMUe. 


T Bundle 
of 63 lbs. 


I Cwt. 

1 


An- 
nealed. 


Bright. 


the aver- 
age den- 
sity of iron 
=1. 




inch. 


milli- 
metres. 


lbs. 


lbs. 


yards. 


yards. , 


lbs. 


lbs. 


average 
iron 
= I. 


7/d 


H 


12.7 


193.4 


3404 


33 


il 


10470 


15700 


I.OI68 


«/o 


^Vsa 


II.9 


170.0 


2991 


37 


66 


9200 


I381O 




5/0 


7/16 


II.I 


148. 1 


2606 


43 


It 


8020 


12000 




4/0 


'3/3a 


10.3 


127.6 


2247 


49 


88 1 


6910 


10370 




3/0 


H 


n 


108.8 


I915 


r 


103 


5890 


8835 


I.O168 


Vo 


"/3a 


91.4 


1609 


69 


123 


4960 


7420 







.326 


8.3 


82.1 


1447 


77 


136 


4450 


6678 




I 


.3a> 


7.6 


69.6 


1227 


^ 


161 


3770 


5655 




2 


.274 


7.0 


5?-^ 


1022 


108 ' 


193 


3140 


4717 




3 


.250 (1) 


6.4 


48.4 


851 


130 


232 


2618 


3927 


I.OI68 


4 


.229 


5.8 


40.6 


714 


*55 


276 1 


2197 


3295 




5 


.209 


5.3 


33.8 


595 


186 


332 


1830 


2740 




6 


.191 


4.9 


28.2 


495 


223 


397 


1528 


2290 




7 


.174 


4.4 


23.4 


412 


269 


479 


1268 


1900 




8 


•159 


4.0 


19.6 


344 


322 


573 


1060 


1558 




9 


.146 


3.7 


16.5 


290 


3^ 


680 


893 


1340 




10 


•133 


3-4 


13.7 


241 


460 


819 


741 


I no 




10^ 


.125 (i) 


3.2 


12. 1 


213 


521 


927 


654 


980 


I.OI79 


II 


.117 


3-0 


10.6 


186 


595 


1059 


573 


860 




12 


.100 (A) 


2.6 


8.0 


142 


783 


1393 


436 


650 




13 


.090 


2.3 


n 


no 


1006 


1790 


339 


509 




14 


.079 


2.0 


!5 


1305 


2322 


261 


390 




IS 


.069 


1.8 


3.7 


65 


1715 
2188 


3052 


199 


299 




16 


.0625(A) 


1.5 


2.9 


51 


3894 


156 


233 


1.0690 


17 


•053 


1-3 


2.2 


38 


2900 


5160 


118 


176 




18 


.047 


1.2 


1.7 


30 


3687 


6560 


93 


138 




19 


.041 


I.O 


1.3 


23 


4847 


8620 


70 


^25 




20 


.036 


.9 


1.0 


18 


5985 


11120 


54 


81 




21 


.03125(A) 


.8 


.8 


H 


7574 


14152 


43 


64 


1. 1765 


22 


.028 


.7 


.6 


II 


9893 


18486 


33 


49 





A/^m. This Table of the weight and strength of Warrington wire is given by permission 
of Messrs. Rylands Brothexs; and it is said to be based on very accurate measurements of 
sires and weights. The last column is added by the author, to show that the density of 
the wire is stationary for diameters of from yi inch to X inch, and probably somewhat 
smaller diameters; but that, contrary to current opinions of the density of wire, the 
density becomes greater when the diameter is reduced to }i inch, and is gradually 
increased as the diameter is further reduced. 



248 WEIGHT OF METALS. 

Table No. 83.— WEIGHT OF WROUGHT-IRON TUBES, 

Bv Internal Diameter. 
Length, i Foot, Thickness by Holuapffel's Wire-GauBe- 



WROUGHT-IRON TUBES. 



249 



Table No. 83 (continued). 
Length, i Foot. Thickness by Holtzapffers Wire-Gauge. 



Thick- 












) 












MCSS. 

W. G. 


8 


9 


10 


II 


12 


13 


14 


15 


16 


17 


18 


Inch. 


.165 


.148 


•134 


.120 


.109 


.095 


.083 


.072 


.065 


.058 


.049 


"/64^- 


9/64/ 


9/64 b. 


%b. 


7/64 


3/32/ 


5/64/ 


s/64^. 


V16/ 


Vi6^. 


3/64/ 


Int. 

DiAM. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


1 
lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


inches. 
























H 


.501 


.423 


.364 


•3*8 


.267 


.219 


.181 


.149 


.130 


.Ill 


.0895 


% 


.717 


.610 


•539 


.472 


.410 


•343 


.290 


.243 


.215 


.187 


•»54 


H 


.934 


.797 


.714 


.625 


.553 


.468 


.398 


.337 


.300 


.263 


.218 


% 


1.15 


1. 00 


.890 


■779 


.695 


.592 


.507 


•431 


.385 


.339 


.282 


H 


1.58 


1-39 


1.24 


1.09 


.981 


.841 


.718 


.620 


•555 


.491 


.410 


1 


2.01 


1.78 


1-59 


1.41 


1.27 


1.09 


.935 


.808 


•Z^5 


.643 


.538 


iH 


2.45 


2.17 


1.94 


1.72 


1.55 


1.34 


I.I5 


.997 


.895 


.795 


.667 


^% 


2.88 


2.55 


2.29 


2.04 


1.84 


1.59 


1-37 


1. 19 


1.07 


.946 


.795 


^M 


3-3' 


2.94 


2.64 


2.35 


2.12 


1.84 


'•59 


1.37 


1.24 


1. 10 


.923 


2 


3.74 


3-33 


3.00 


2.66 


2.41 


2.08 


1. 81 


1.56 


1.41 


1.25 


1.05 


2% 


4.17 


3.72 


3-35 


2.98 


2.69 


2.33 


2.02 


1.75 


1.58 


1.40 


1. 18 


2% 


4.61 


4.10 


3.70 


329 


2.98 


2.58 


2.24 


1.94 


1.75 


1.55 


1.31 


2^ 


504 


^2S 


4.05 


3.61 


3.26 


2.83 


2.46 


2.13 


1.92 


Hi 


1.44 


3 ^ 


5.47 


4.88 


4.40 


392 


3.55 


3.08 


2.68 


2.31 


2.09 


1.86 


'•57 


Z}i 


6.33 


5.65 


5.10 


^•55 


4.12 


3.58 


3" 


2.69 


2.43 


2.16 


1.82 


4 


7.20 


6.43 


5.80 


5.18 


4.69 


4.07 


3.55 


307 


2.77 


2.47 


2.08 


4K 


8.06 


7.20 


6.50 


5.81 


5.26 


4.57 


3.98 


3-45 


3." 


2.77 


'2.34 


5 


8.93 


7.98 


7.21 


6.44 


5.83 


5.07 


4.42 


3.83 


3.45 


3.07 


2.59 


\^ 


9.79 


8.75 


7.91 


7.06 


6.40 


5.57 


4.85 


4.20 


3.79 


3|! 


2.85 


6 


10.7 


9.53 


8.61 


7.69 


6.97 


6.07 


5.29 


4.58 


4. 13 


3.68 


3. 1 1 


1 6j^ 


11.5 


10.3 


9.31 


8.32 


7.55 


6.56 


§•72 


4.96 


4.47 


3.98 


3.36 


' 7 


12.4 


ii.i 


10. 


8.95 


8.12 


7.06 


6.16 


5-33 


4.81 


4.29 


3.62 


l^ 


133 


II. 9 


10.7 


958 


8.69 


Z-55 


6.59 


5.71 


5.15 


4.59 


3.88 


8 


14. 1 


12.6 


11.4 


10.2 


9.26 


8.06 


7.03 


6.09 


5-49 


4.90 


4.13 


9 


15.8 


14.2 


12.8 


II. 5 


10.4 


9.05 


7.90 


6.84 


6.17 


5.50 


4.65 


10 


17.6 


»5.7 


14.2 


12.7 


II.5 


lO.O 


8.77 


7.60 


6.85 


6.11 


5.16 


II 


19.3 


^7-3 


15.6 


14.0 


12.7 


II. 


964 


8.35 


7-53 


6.72 


5.67 


12 


21.0 


18.8 


17.0 


15.2 


13.8 


12.0 


10.5 


9.10 


8.21 


7.33 


6.19 


13 


22,7 


20.4 


18.4 


16.5 


15.0 


130 


11.4 


9.86 


8.89 


7-93 


6.70 


14 


245 


21.9 


19.8 


17.7 


16. 1 


14.0 


12.2 


10.6 


9.57 


8.54 


7.22 


'5 


26.2 


23-5 


21.3 


19.0 


17.2 


15.0 


13.1 


11.4 


10.3 


Kl 


7.73 


16 

1 


27.9 


25.0 


22.7 


20.3 


18.4 


16.0 


14.0 


12. 1 


10.9 


9.88 


8.24 


I '7 


29.6 


26.6 


24.1 


21.5 


19.5 


17.0 


14.9 


12.9 


11.6 


10.4 


8.76 


' 18 

* 


3»-4 


28,1 


255 


22.8 


20.6 


; 18.0 


15.7 


13.6 


12.3 


II. 


9.27 


;i9 


33.1 


29.7 


26.9 


24.0 


21.8 


1 19.0 


16.6 


14.4 


130 


11.6 


9.78 


• 20 


34-8 


31.2 


28.3 


25-3 


22.9 


20.0 


^Z-5 


15.1 


13.7 


12.2 


10.3 
10.8 


21 


36.6 


32.8 


29.7 


26.? 


24.1 


21.0 


18.3 


^H 


14.3 


12.8 


22 


38.3 


34.3 


311 


27.8 


25.2 


1 22.0 


19.2 


16.6 


15.0 


13.4 


II. 3 


23 


40.0 


35.9 


32.5 


29.1 


26.4 


1 23.0 


20.1 


17.4 


15.7 


14.0 


II. 8 


24 


41.8 


37.4 


33-9 


303 


27.5 


1 24.0 


20.9 


18. 1 


16.4 


14.6 


12.6 


26 


45.2 
48.7 


40.5 


367 


32.8 


29.8 


1 26.0 


22.6 


197 


17.7 


15.8 


134 


28 


43-6 


39.5 


35.3 


32.1 


28.0 


24.4 


21.2 


19. 1 


17.0 


14.4 


30 


52.1 


46.7 


42.3 


37.8 


34.4 


30.0 


26.1 


22.7 


20.5 


18.3 


15.4 


32 


55.5 


49.8 


48.0 


40.4 


36.7 


32.0 


27.9 


24.2 


21.8 


19.5 


16.5 


34 


59.0 


52.9 


42.9 


39.0 


34.0 


29,7 


25.8 


23.2 


20.7 


^I'i 


>36 


62.4 


56.0 


50.8 


45-4 


41.3 


1 36.0 


31-4 


27.3 


24.6 


21.9 


18.6 



250 



WEIGHT OF METALS. 



Table No. 84.— WEIGHT OF WROUGHT-IRON TUBES, 

BY External Diameter. 

Length, i Foot. Thickness by Holtzapffel's Wire- Gauge. 



Thickness. 




















W. G. 


7 


8 


9 


ID 


II 


la 


13 


14 


15 


Inch. 


.180 


.165 


.148 


.134 


.120 


.109 


.095 


.083 


.072 




3/x6 ^. 


"/64 ^• 


9/64/ 


9/64 ^. 


yib- 


7/64 


3/33/ 


5/64/ 


5/64 b. 


Ext. Diam. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


I inch. 


'•55 


1.44 


1-32 


1.22 


I. II 


1.02 


.900 


.797 


.700 


'H 


1.78 


1.66 


I.5I 


1-39 


1.26 


1. 16 


1.03 


.906 


•794 


'^ 


2.02 


1.88 


I.7I 


I.S7 


1.42 


1.30 


1. 15 1 I.OI 


.888 


'H 


2.25 


2.09 


1.90 


1.74 


1.58 


1.45 


1.27 ! 1. 12 


.983 


'^ 


2.49 


2.31 


2.10 


1.92 


^'P 


1.59 


1.40 ! 1.23 


1.08 


iH 


2.72 


2.52 


2.29 


2.09 


1.89 


H'^ 


1.52 1 1.34 


1.17 


'H 


2.96 


2.74 


2.48 


2.27 


2.05 


1.87 


1.65 , 1.45 


1.27 


m 


3.19 


2.96 


2.68 


2.45 


2.21 


2.02 


1.77 I 1.56 


1.36 


2 


3-43 


3.17 


2.87 


2.62 


2.36 


2.16 


1.90 1 1.67 


1-45 


i% 


3.67 


3-39 


3.06 


2.80 


2.52 


2.30 


2.02 


1.78 


1-55 


^% 


3.90 


3.60 


326 


2.97 


2.68 


2.44 


2.14 


1.88 


1.64 


2H 


4.14 


3.82 


3.45 


3.15 


2.83 


2.59 


2.27 


1.99 


1.74 


*^ 


4.37 


4.04 3.65 


332 


2.99 


2.73 


2.39 


2.10 


1.83 


2H 


4.61 


4.25 


3.84 


350 


3.15 


2.87 


2.52 


2.21 


1-93 


2¥ 


4.84 


4.47 


4.03 


3.67 


3-31 


3.02 


2.64 


2.32 


2.02 


2^ 


5.08 


4.68 


4.23 


3.85 


3-46 


3.16 


2.77 


2.43 


2.11 


3 , 


5.32 


4.90 


4.42 


4.02 


3.62 


3.30 


2.89 


2.54 


2.21 


yx 


VI 


5.33 


4.81 


4-37 


3-94 


3-59 


314 


2.75 


2.40 


i'A 


6.26 


5.76 


5.20 


4.72 


4.25 


387 


3.39 


2.97 


2.59 


2K 


6.73 


6.19 


5.58 


5-07 


4.88 


4.16 


3.64 


3.19 


2.77 


4 


7.20 


6.63 


5.97 


5.43 


4.44 


3.89 


340 


2.96 


A% 


7.67 


7.06 6.36 


5.78 


5.20 


4.73 


4.38 


3.62 


3.15 


*>i 


8.14 


7-49 


7.45 


6.13 


5-5' 


5.01 


3.84 


3-34 


4^ 


8.61 


7.91 


713 


6.48 


5.82 


5-30 


4.63 


4.06 


3-53 


s , 


9.08 


8.35 


7.52 


6.83 


6.13 


5.58 


4.88 


4.27 


372 


5^ 


9.56 


8.79 


7.91 
8.30 


7.18 


6.44 


5-»7 


5.13 


4.49 


3-90 


s'4 


10.0 


9.22 


7.88 


6.76 


6.15 


5-38 


4.71 


4.09 


k^ 


10. 5 


9.65 


8.68 


7.07 


6.44 


5.63 


4.93 


4.28 


6 


II.O 


ZO.I 


9.07 


8.23 


7.39 


6.73 


5.87 


5.14 
5.36 


4.47 


6H 


11.4 


10.5 


9.46 


8.58 


7.70 
8.02 


7.01 


6.12 


4.66 


(,'A 


11.9 


10.9 


9.85 


8.93 


7.30 


6.37 5-58 


4.8s 


(>K 


12.4 


11.4 


10.2 


9.28 


8.33 


7.58 


6.62 


5-79 


503 


7 


12.9 


1 1.8 10.6 


9.63 


8.64 


7.87 


6.87 


6.01 


5.22 


1% 


'3-3 


12.2 II.O 


9.99 


8.96 


8.15 


7.12 


6.23 


541 


1% 


13.8 


12.7 


11.4 


10.3 


9.27 


8.44 


7.37 


6.45 


5.60 


l^ 


14.3 


131 


II. 8 


10.7 


9-59 


8.72 


7.62 


6.66 


5-79 


8 


14.7 


13.5 


12.2 


II.O 


9.90 


9.01 


7.86 


6.88 


5.98 


Thicknkss. 






















W. G. 






4 


5 




6 




7 


8 


9 


Inch. 


•3125 


.281 


.238 


.220 


.2 


J03 


* 


180 


.165 


.148 




5/16 


9/3a 


^5/64/ 


7/33 


r X3 


'/64 


^ 


Ueb. 


"/64 ^- 


9/64/ 


Ext. Diam. 


lbs. 


lbs. 


lbs. 


lbs. 


"7 


bs. 


1 


lbs. 


lbs. 


lbs. 


7 inch. 


21.9 


19.8 


16.9 


15.6 


li 


^5 




2.9 


II. 8 


10.6 


l^ 


23-5 


21.3 


18. 1 


16.8 


II 


>•§ 




3.8 


12.7 


II.4 


8 


25.2 


22.7 


19.3 


17.9 


i( 


3.6 




4.7 


'35 


12.2 


8^ 


26.8 


24.2 


20.6 


19. 1 


i: 


r.6 




5-7 


14.4 


12.9 


9 


28.4 


25-7 


21.8 


20.2 


i\ 


H 




6.6 


15.3 


137 


9>^ 


30.1 


27.1 ! 23.1 


21.4 


i< 


?.8 




7.6 


16. 1 


14.5 


10 


31.7 


28.6 ! 24.3 


22.5 


1 ^ 


>.8 




8.5 


17.0 


15.3 



LIST OF TABLES OF CAST IRON, STEEL, ETC. 2$ I 



LIST OF TABLES OF THE WEIGHT OF CAST IRON, 
STEEL, COPPER, BRASS, TIN, LEAD, AND ZINC. 

The following Tables are devoted to the specialities of manufacture in 
Cast Iron, Steel, and other metals, embracing the utmost range of dimen- 
sions to which objects in the several metals are executed in the ordinary 
course of practice. 

Thus, whilst it is customary for certain classes of Cylinders in Cast Iron — 
steam cylinders, for example — to be constructed according to given internal 
diameters, other classes are constructed according to diameters given 
externally, as the iron piers of railway bridges. Two distinct tables accord- 
ingly have been composed, showing the weights of Cylinders of various 
thicknesses, and of diameters as measured internally and externally. 

The weights of Copper Pipes and Cylinders are only calculated for in- 
ternal diameters, as it is not the practice to construct them to given external 
diameters. Brass Tubes, on the contrary, are calculated only for external 
diameters, as they are not ordinarily made to given internal diameters. 

Table No. 85. — ^Weight of Cast-iron Cylinders, i foot in length, advanc- 
ing, by internal measurement, from i inch to 10 feet in diameter, and from 
}^ inch to 2j^ inches in thickness. 

Table No. 86. — ^Weight of Cast-iron Cylinders, i foot in length, advanc- 
ing, by external measurement, from 3 inches to 20 feet in diameter, and 
from ^Ixt inch to 4 inches in thickness. 

Table No. 87. — Volume and weight of Cast-iron Balls, when the 
diameter is given; from i inch to 32 inches in diameter, with multipliers 
for other metals. 

Table No. 88. — Diameter of Cast-iron Balls, when the weight is given ; 
from yi pound to 40 cwts. 

Table No. 89. — ^Weight of Flat Bar Steel, i foot in length ; from ^ inch 
to I inch thick, and from }i inch to 8 inches in width. 

Table No. 90. — Weight of Square Steel, i foot in length ; from }i inch 
to 6 inches square. 

Table No. 91. — ^Weight of Round Steel, 1 foot in length; from }i inch 
to 24 inches in diameter. 

Table No. 92. — ^Weight of Chisel Steel: hexagonal and octagonal, i foot 
in length; from ^ inch to i^ inches diameter across the sides. 
Oval-flat, from J^ x ^ inch to ij5^ x ^ inch. 

Table No. 93. — ^Weight of one square foot of Sheet Copper; from No. i 
to No. 30 wire-gauge, as employed by Williams, Foster, & Co. 

Table No. 94. — Weight of Copper Pipes and Cylinders, i foot in length, 
advancing, l)y internal measurement, from }i inch to 36 inches in diameter, 
and from No. 0000 to No. 20 wire-gauge in thickness. 



252 WEIGHT OF METALS. 

Table No. 95. — ^Weight of Brass Tubes, i foot in length, advancing, by 
external measurement, from }i inch to 6 inches in diameter, and from 
No. 3 to No. 25 wire-gauge in thickness. 

Table No. 96. — ^Weight of one square foot of Sheet Brass; from No. 3 to 
No. 25 wire-gauge in thickness. 

Table No. 97. — Size and weight of Tin Plates. 

Table No. 98. — ^Weight of Tin Pipes, as manufactured. 

Table No. 99. — Weight of Lead Pipes, as manufactured. 

Table No. 100. — Dimensions and weight of Sheet Zinc. (VicUe-Mon- 
tagne,) 



CAST-IRON CYLINDERS. 



253 



Table No. 85. — Weight of CastJron Cylinders. 



By Internal Diameter, i Foot Long. 



Int. 


Thickness in Inches. 


DiAM. 


























X 


5/16 


H 


7/16 


>i 


9/16 


H 


"/I6 


H 


H 


I 


inches. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


I 


307 


4.03 


5.06 


6.17 


7.36 


8.63 


9.97 


II.4 


12.9 


16. 1 


19.6 


^'A 


4.30 


5.56 


6.90 


8.32 


9.82 


II.4 


13. 1 


14.8 


16.6 


20.4 


245 


2 


5.52 


7.09 


8.74 


10.5 


12.3 


14.2 


16. 1 


18. 1 


20.3 


247 


29.5 


2>i 


6.75 


8.63 


10.6 


12.6 


147 


16.9 


19.2 


21.5 


23-9 


29.0 


344 


3 , 


7.9« 


10.2 


12.4 


14.8 


17.2 


19.7 


22.2 


24.9 


27.6 


33.3 


39.3 


3>i 


9.20 


II.7 


143 


16.9 


19.6 


22.4 


25.3 


28.3 


3'.3 


37.6 


44.2 


4 ^ 


10.4 


13.2 


16.1 


19. 1 


22.1 


2Q.2 
28.0 


28.4 


31.6 


350 


41.9 


49.1 


4^ 


II.7 


14.8 


18.0 


22.1 


245 


31.5 


35-0 


38.7 


46.2 


54.0 


5 , 


12.9 


16.5 


19.8 


23.4 


27.0 


307 


345 


38.4 


42.3 


50.5 


1^-2 


5^ 


14-1 


17.8 


21.6 


255 


29.5 


33.5 


37.6 


41.8 


46.0 


54.8 


ti'^ 


^ 


15.3 


19.4 


235 


27.7 


32.0 


36.2 


40.7 


45.1 


49.7 


59.1 


68.7 




Thickness in Inches. 


1 


H 


7/x6 


H 


9/x6 


H 


"/»6 


H 


H 


I 


IH 


iH 


inches. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lU. 


lbs. 


6 


23.5 


27.7 


32-0 


36.2 


40.7 


45.1 


49.7 


59-1 


68.7 


78.7 


89.0 


^H 


25.3 


29.8 


344 


390 


^3-7 


48.5 


53.4 


63.4 


73.6 


84.2 


95-1 


7 , 


27.2 


32.0 


36.8 


41.8 


46.8 


51.9 


57-; 


67.7 




89.7 


101.2 


7}i 


29.0 


34.1 


39.3 


44.5 


49.9 


55.3 


60.8 


71.9 


95.3 


107.4 


8 


30.8 


36.3 


41.7 


47.3 


52.9 


58.6 


64.4 


76.2 
80.5 


100.8 


"35 


8)^ 


32.7 


38.4 


44.2 


50.0 


55.9 


62.0 


68.1 


93-3 


106.3 


119. 7 


' ^. 


34.5 


40.5 


46.6 


52.8 


59.0 


U.t 


71.8 


84.8 


98.2 


III.8 


125.8 


. 9>^ 


36.4 


42.7 


49.1 


55.6 

58.3 
61. 1 


62.0 


75.5 


89.1 


lOJI 

108.0 
112.9 


117.4 


131.9 


• 10 
1 «o>^ 


3».2 
40.0 


44.8 
47.0 


51.5 
54.0 


65.1 
68.2 


72.1 

75-5 


82.8 


93.4 
97.7 


122.9 
128.4 


138. 1 
144.2 


' 11 


41.9 


49.1 


56.5 


63.9 


71.2 


78.9 


86.5 


102.0 


1 17.8 


133.9 


150.3 


11;^ 


43-7 


51.3 


58.9 


66.6 


745 


82.3 


90.2 


106.3 


122.7 


139.4 


156.5 


12 


45.6 


53.4 


61.4 


69.4 


83.6 


85.6 


939 


no. 6 


127.6 


145.0 


162.6 


1 13 


49.2 


57.7 


66.3 


749 
80.4 


92.4 


101.2 


1 19.2 


137.5 


156.0 


1749 
187.2 


. ^^ 


52.9 


62.0 


71.2 


89.7 


99.1 


108.6 


127.8 


147.3 


167. 1 
178.1 


1 15 


^1 


66.3 


76.1 


85.9 


95-9 


105.9 


116.0 


136.4 


157. 1 


199.4 


16 


70.6 


81.0 


91.5 


102.0 


112.6 


123.3 


145.0 


166.9 


189. 1 


211.7 


17 


64.0 


749 


85.9 


97.0 


108.2 


1 19. 4 


130.7 


153-6 


176.7 
186.5 


200.2 


224.0 


18 


67.7 


79.2 


|9o.8 


102.5 


"43 


126. 1 


138. 1 


162.2 


211. 2 


236.2 




Thickness in Inches. 


H 


7/16 


H 


H 


H 


H 


I 


iH 


iX 


m 


i^ 


inches. 


cwt. 


cwt. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


' 18 


.604 


.707 


.811 


1.02 


1.23 


1-45 


1.67 


1.89 


2.II 


2.34 

2.46 


2.56 


19 


.637 


.746 


:S 


1.08 


1.30 


1.52 


1.75 


1.99 


2.22 


2.70 


20 


.670 


.784 


113 


1.36 


1.60 


1.84 


2.08 


2.33 


2.58 


2.83 


21 


.703 


.823 


.942 


1. 19 


1-43 


1.68 


1.93 


2.18 


2.44 


2.70 


2.96 


22 


.736 


.861 


.986 


1.24 


1.49 


1.76 


2.02 


2.28 


2.55 


2.82 


309 


' 23 


.769 


.900 


1.03 


1.29 


1.56 


1.83 


2.10 


2.38 


2.66 


2.94 


3-22 


24 


.802 


.939 


1.07 


1.35 


1.63 


1.91 


2.19 


2.48 


HI 


3.06 


3-35 


25 


.835 


.977 


1. 12 


1.40 


1.69 


1.99 


2.28 


2.58 


2.88 


3.18 


3-48 



254 



WEIGHT OF METALS. 



Table No. 85 {continued). 



By Internal Diameter, i Foot Long. 



Int. 










Thickness in 


Inches. 










DiAM. 


























H 


7/x6 


% 


H 


H 


n 


I 


1% 


1% 


m 


i}i 


inches. 


CWU. 


cwts. 


cwts. 


cwts. 


t 

cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


26 


.868 


1.02 


1. 16 


1.46 


1.76 


2.06 


2.37 


2.68 


2.99 


330 


362 


27 


.901 


1.05 


I.2I 


I.5I 


1.82 


2.14 


2.45 


2.77 


309 


3.42 


HI 


28 


.934 


1.09 


1.25 


1-57 


1.89 


2.22 


2.54 


2.87 


3.20 


3.54 


3.8S 


29 


.967 


i.i-:« 


1.29 


1.62 


1.96 


2.29 


2.63 


2.97 


3.31 


3.66 


4.01 


30 


.998 


1.17 


1.34 


1.68 


2.02 


2.37 


2.72 


3.07 


342 


3-78 


4.14 


32 


1.06 


1.25 


1-43 


1.79 


2.15 


2.52 


2.89 


327 


364 


4.02 


4.41 


3i 


1.13 


1.32 


1.51 


1.90 


2.29 


2.67 


3.07 


346 


3.86 


4.26 


4.67 


36 


1.20 


1.40 


1.60 


2.01 


2.42 


2.83 


3.24 


3.66 


4.08 


4-50 


4.94 


38 


1.26 


1.47 


1.69 


2.12 


^•§§ 


2.98 


3.42 


3.86 


4.30 


4.75 


5.20 


40 


1.33 


1.55 


1.77 


2.23 


2.68 


314 


3.59 


4.05 


4.52 


4.99 


5.47 


42 


1.39 


1.63 


1.86 


2.34 


2.81 


3.29 


3.77 


4.25 


4.74 


5-23 


5.73 


45 


1.49 


1.75 


1.99 


2.50 


3.01 


3.52 


4.03 


4.55 


5.07 


5-5? 


6.13 


48 


1.59 


1.86 


2.12 


2.66 


3-21 


3-75 


430 


4.85 


5.40 


5.96 


6.52 












Thick 


NBSS IN 


Inches. 










H 


H 


Ji 


1 


>X 


1% 


iji 


i>i 


I^ 


2 


2X 


inches. 


cwts. 


cwts* 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


48 


2.66 


3.21 


3-75 


430 


4.85 


5.40 


5.96 


6.52 


7.63 


8.77 


9.91 


SI 


2.82 


3.40 


3.98 


4.56 


S'H 


5-73 


6.32 


6.91 


8.09 


9.29 


10.5 


54 


2.99 


3.60 


4.21 


4.82 


5-44 


6.06 


6.69 


7.31 


8.55 


9.82 


II.I 


57 


3.15 


3.80 


4-44 


5.09 


5.73 
6.03 


6.38 


7.05 


7.70 

8.10 


9.01 


10.4 


II.7 


60 


3.32 


4.00 


4.67 


5-35 


6.71 


7.41 


9-47 


10.9 


12.3 


63 


3.48 


4.19 


4.90 


5.61 


6.33 


7.04 


7.78 


8.49 


9-93 


II.4 


12.9 


66 


3.64 


4-39 


5.13 


5.88 


6.62 


7.37 


8.14 


8.89 


10.4 


II.9 


13-5 


69 


3.81 


4.59 


5.36 


6.14 


6.92 


7.70 


8.51 


9.28 


10.9 


12.5 


14.1 


72 


3.97 


4.78 


5- §9 


6.40 


7.21 


8.03 


8.87 


9.67 


"•? 


130 


14.7 


75 


4.14 


4.98 


5.82 


6.66 


7-5' 


8.36 


9.24 


10. 1 


11.8 


13.5 


15.2 


78 


4.30 


5-^! 


6.05 


6.93 


7.81 


8.69 


9.60 


10.5 


12.2 


14.0 


15.8 


81 


4.46 


5.38 


6.28 


7-19, 


8.10 


9.02 


9.97 


10.9 


12.7 


14.6 


16.4 


84 


4.63 


5-57 


6.51 


7.45* 


8.40 


9.35 


10.3 


"3 


13.2 


I5.I 


17.0 


87 


4.79 


5.77 


6.74 


7.72 


8.69 


9.67 


10.7 


II. 6 


136 


15.6 


17.6 


90 


4.96 


5-97 


6.97 


7.98 
S.24 


8.99 


10.0 


II.I 


12.0 


14. 1 


16. 1 


18.2 


93 


5.12 


6.17 


7.20 


9.29 


10.3 


1 1.4 


12.4 


14.5 


16.7 


18.8 


96 


5.28 


6.36 


HI 


8.51 


9.58 
9.88 


10.7 


II. 8 


12.8 


15.0 


17.2 


19.4 


99 


5-45 


6.56 


7.66 


8.77 


II.O 


12.2 


13.2 


15.5 


17-7 


20.0 


102 


5-^i 


6.76 


7.89 


903 


10.2 


"3 


12.5 


13-6 


15.9 


18.2 


20.6 


loq 


5.78 


6.95 


8.12 


9.29 


10.5 


II. 7 


12.9 


14.0 


16.4 


18.8 


21.2 


108 


5.94 


7.15 


8.36 


9.56 


10.8 


12.0 


13-3 


14.4 


16.8 


'9-3 


21.8 


III 


6.10 


7.35 


f-S9 


9.82 


ii.i 


12.3 


136 


14.8 


17.8 


19.8 


22.3 


114 


6.27 


7.55 


8.82 


10. 1 


11.4 


12.6 


14.0 


15.2 


20.3 


22.9 


117 


6.43 


7.74 


9.05 


10.4 


11.7 


13.0 


14.3 


K.6 
iS.o 


18.2 


20.9 


23-5 


120 


6.59 


7-94 


9.28 


10.6 


12.0 


13.3 


14.7 


18.7 


21.4 


24.1 



CAST-IRON CYLINDERS. 



^SS 



Table No. 86. — Weight of Cast-Iron Cylinders. 
By External Diameter, i Foot Long. 



Ext. 










Thickness in 


Inches. 








DiAM. 






1 




















3/16 


X 


5/x6 ' 

I 


H 


7/16 


K 


9/16 


H 


U 


H 


I 


inches. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


3 


5.18 


6.75 
7.98 


8.25, 


9.65 


II.O 


12.3 


13.5 


14.6 


16.6 


18.3 


19.6 


Z}^ 


6.10 


9.78! 


11.5 


13.2 


14.7 


16.2 


17.6 


20.3 


22.6 


24.5 


4 , 


7.02 


9.20 


"3 


13.3 


15.3 


17.2 


19.0 


20.7 

23.8 


24.0 


26.9 


29.5 


4,^ 


7-94 


10.4 


12.9 


15.2 


17.5 


19.6 


21.7 


27.7 


31.1 


344 


5 , 


8.86 


II.7 


14.4 


17.0 


19.6 


22.1 


24.5 


26.9 


31.5 


35.4 


39.3 


5^ 


9.78 


12.9 


15.9 


18.9 


21.8 


24.5 


27.3 


29.9 


35-2 


39.7 


44.2 


6 


10.7 


14. 1 


17.5 


20.7 


23.9 


27.0 


30.0 


33.0 


38.9 


44.0 


49.1 


6X 


II. 6 


*$•? 


19.0 


22.5 


26.0 


29.5 


32.8 


36.1 


42.6 


48.3 


54.0 


7 


12.5 


16.6 


20.5 


24.4 


28.2 


31.9 


35-6 


39.1 


46.4 


52.6 


58.9 


l}i 


13.5 


17.8 


22.1 


26.2 


30.3 


34.4 


38.3 


42.2 


5°i 


56.9 


63.8 
68.7 


8 


14.4 


19.0 


23.6 


28.1 


32.5 


36.8 


41.1 


^§•3 


53-8 


61.2 


8^ 


153 


20.3 


25-1 


29.9 


34.6 


39-3 


43-8 


48.3 


57.5 


^J'i 


73.6 


9 


16.2 


21.5 


26.7 


31.8 


36.8 


41.7 


46.6 


51.4 


61.3 


69.8 




9H 
lo 


17.2 
18.1 


22.7 
23.9 


28. 2 
29.7 


33.6 

35-4 


38.9 

41. 1 


44.2 
46.6 


49.4 
52.1 


54.5 
57.5 


65.0 
68.7 


78.4 


II 


19.9 


26.4 


32.8 


39.1 


45-4 


51.5 


57.6 


63.7 


76.0 


87.0 


98.2 


12 


21.8 


28.8 


35.9 


42.8 


49.7 


56.5 


63.2 


69.8 


83.4 


95.6 


108.0 


13 


23.6 


31.3 


38.9 


46.5 


54.0 


61.4 


68.7 


75.9 


90.7 


104.2 


117.8 


14 


255 


33*8 


42.0 


50.2 


58.3 


66.3 


74.2 


82.1 


98.0 


112.8 


127.6 


15 


27.3 


36.2 


45-* 


53.8 


62.6 


71.2 


79.7 
85.3 


88.2 


105.4 


121.3 


137.4 


i6 


29.1 


38.7 


48.1 


57.5 


66.9 


76.1 


94.3 


112.7 


129.9 


147.3 


17 


3'° 


41. 1 


51.2 


61.2 


71.1 


81.0 


90.8 


100.5 


120.0 


138.5 


157.1 


i8 


32.8 


43-6 


54.3 


64.9 


75.4 


85.9 


96.3 


106.6 


127.4 


147. 1 


166.9 


19 


34-6 


46.0 


57.3 


68.6 


79-7 


90.8 


101.8 


112.8 


1347 


155.7 


176.7 


. 20 

1 


36.5 


48.5 


60.4 


72.3 


84.0 


95.7 


107.3 


118.9 


142.0 


164.3 


186.5 


' 21 


38.3 


50.9 


63.5 


75.9 


88.3 


100.6 


112.9 


125.0 


149.4 


172.9 


196.4 


1 22 


40.2 


53-1 


66.5 


79.6 


92.6 


105.5 


118.4 


131.2 


156.7 


181. 5 


206.2 


23 


42.0 


55.8 


69.6 


83.3 


96.9 


"0.5 


123.9 


1373 


164.0 


190.1 


^^5;0 


i 24 


43.8 


f^§ 


72.7 


87.0 


101.2 


115.4 


129.4 


143-4 


171.4 


198.7 


225:8 


. 25 


45-7 


60.8 


7S-Z 


90.7 


105.5 
109.8 


120.3 


135.0 


149.6 


178.7 


207.2 


235.6 


26 


47.5 


63.2 


78.8 


943 


125.2 


140.5 


'§5-2 


186.1 


215.8 


245.4 


27 


49.4 


^§•7 


81.9 


98.0 


114.1 


130. 1 


146.0 


161.8 


193.4 


224.4 


2553 


28 


51-2 


68.1 


85.0 


101.7 


118.4 


135.0 


151.5 


168.0 


200.7 


233.0 


265.1 


29 


53-0 


70.6 


88.0 


105.4 


122.7 


139.9 


157.0 


174.1 


208.1 


241.6 


274.9 


30 


54-9 


730 


91. 1 


109. 1 


127.0 


144.8 


162.6 


180.2 


215.4 


250.2 


284.7 


31 


56.7 


75-5 


94.2 


112.8 


131.3 


149.7 


168.1 


186.4 


222.7 


258.8 


294.5 


: 32 


58.6 


77.9 


97.2 


1 16.4 


135.6 


1546 


173.6 


192.5 


230.1 


267.4 


304.3 


33 


60.4 


80.4 


100.3 


1 20. 1 


1399 


159.5 


179.1 


198.7 
204.8 


237.5 


276.0 


314.2 


|34 


62.2 


82.8 


103.4 


123.8 


144.2 


164.5 


184.7 


244.8 


284.6 


324.0 


35 


64.1 


IH 


106.4 


127.5 


148.5 


169.4 


190.2 


210.9 


252.2 


293.1 


333.8 


36 


65.9 


87.8 


109.5 


131.2 


152.7 


174.3 


1957 


217.1 


259.5 


301.7 


343.6 


38 


69.6 


92.7 


115.6 


138.5 


161.3 


184. 1 


206.8 


229.3 


274.3 


3189 


363.2 


40 


73-3 


97.6 


121.8 


145.9 


169.9 


193.9 


217.8 


241.6 


289.0 


336.1 


382.9 


42 


77.0 


102.5 


127.9 


153.3 


178.5 


203.7 


228.8 


253.9 


3037 


353.3 


402.5 


Jl 


82-5 


109.8 


137. 1 


164.3 


191.2 


218.5 


245.4 


272.3 


325.8 


379.1 


432.0 


88.0 


117.2 


146.3 


175.4 


203.8 


233.2 


262.0 


290.7 


347.9 


404.8 


461.4 


5« 


93.6 


124.6 


155-5 


186.4 


216.5 


247.9 


278.6 


309.1 


370.0 


430.6 


490.9 


54 


99.1 


131.9 


164,7 


197.5 


229.2 


262.6 


295.1 


327.5 


392.1 


456.4 


520. J 


57 


104.6 


139.3 


173.9 


208.5 


241.8 


277.4 


311.7 


345.9 


414.2 


482.1 


549.8 


60 


IIOlI 


146.6 


183. 1 

• 


219.6 


254.5 


292.1 


328.3 


364.3 


436.3 


507.9 579.3 



256 



WEIGHT OF METALS. 



Table No. 86 {continued), 
Bv External Diameter, i Foot Long. 



Ext. 








1 

Thicknrss in Inchbs. 


DiAM. 


























3/i6 


% 


5/16 


H 


7/16 


>i 


9/16 


H 


^ 


Ji 


z 


ft. in. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


53 


1.03 
1.08 


1.44 


1.71 


2.06 


2.39 


2.74 


3.08 


3.42 


4.09 


4.77 


5-43 


56 


1.50 


1.80 


2.16 


2.50 


2.87 


3.22 


3.58 


4.29 


5.00 


5-70 


59 


'•'2 


1.55 


1.88 


2.26 


2.62 


300 


3-37 


3.75 


4-49 


523 


5.96 


60 


1. 18 


1.61 


1.96 


2.36 


2.74 


3- 14 


3.52 


3.91 


4-69 


546 


6.22 


63 


1.23 


1.67 


2.05 


2.45 


2.85 


327 


3.66 


4.08 


4.88 


5.69 


6.49 


66 


1.28 


1.73 


2.13 


2.55 


2.97 


340 


3.81 


4.24 


5.08 


5.92 


6.75 


69 


'•33 


1.78 


2.21 


2.65 


3.09 


3-53 


3.96 


4.41 


5.28 


6.38 


7.01 


70 


1.38 


1.84 


2.29 


2.75 


3.20 


3.66 


4.10 


4.57 


5-57 


7.28 


76 


1.48 


1.95 


2.46 


2.95 


3.43 


3.92 


4.39 


4.90 


5.87 


6.84 


7.80 


80 


1.58 


2.07 


2.62 


3.15 


3.67 


4.19 


4.69 


5.23 


6.26 


7.30 


8.33 










Thick 


NBSS IN Inches. 


^% 


iX 


1^8 


^% 


im: 


2 
{ cwts. 


2X 


2>4 


2^ 


3 


3>^ 


4 


inches. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


cwts. 


6 


.481 


.520 


.557 


.592 


.652 


.701 


'W. 


.761 










6>i 


•530 


•575 


.618 


.657 


.729 


.789 


.838 


.872 


.906 








7 


•579 


.630 


.678 


.723 


.805 


.876 


.938 


.982 


1.03 


1.05 






VA 


.629 


.68s 


.738 


.789 


.882 


.964 

1 


1.04 


1.09 


I.I5 


1. 18 






8 


.678 


.740 


.799 


.855 


.959 


i.os 


1. 14 


1.20 


1.27 


1.32 


1.38 




8>^ 


•727 


.794 


.859 


•^11 


1.04 


1. 14 


1.23 


I.3I 


1.39 


'•45 


1.53 




9 


.777 


.849 


.919 


.986 


I. II 


1.23 


1-33 


1.42 


I.5I 


1.58 


1.69 


1.75 


9y2 


.826 


.904 


.980 


1.05 


I.I9 


1 M' 


1.43 


1.53 


1.63 


I.7I 


1.84 


1.93 


10 


.875 


.959 


1.04 


1. 12 


1.27 


1.40 


1.53 


1.64 


1.75 


1.84 


1.99 


2.10 


II 


.974 


1.07 


i.x6 


X.25 


1.42 


1.58 


1.73 


1.86 


1-99 


2.10 


2.30 


2.46 


12 


1.07 


I.18 


1.28 


1.38 


1.57 


1 1.75 


1.92 


2.08 


2.23 


2.37 


2.61 


2.81 


13 


1.17 


1.29 


1.40 


1.51 


1-73 


1.93 


2.12 


2.30 


2.47 


2.63 


2.92 


3.16 


H 


1.27 


1.40 


1.52 


1.64 


1.88 


2.10 


2.32 


2.52 


2.71 


2.89 


3.22 


l& 


15 


1.37 


1.51 


1.65 


1.78 


2.03 


2.28 


2.52 


2.74 


2.95 


3.16 


\^ 


16 


1.47 


1.62 


1.77 


I.9I 


2.19 


2.45 


2.71 


2.96 


319 


3-12 


4.21 


17 


1.57 


1.73 


1.89 


2.04 


2.34 


2.63 


2.91 


3.18 


3.44 


3.68 


4.14 


4.56 


18 


1.66 


1.84 


2.01 


2.17 


2.49 


2.81 


3." 


3.40 


3.68 


3-95 


4.45 


4.91 


20 


1.86 


2.06 


2.25 


2.43 


2.80 


3.16 


3-50 


3.83 


4.16 


4.47 


5-^ 


5.61 


22 


2.06 


2.27 


2.49 


2.70 


3." 


l^ 


3.90 


4.27 


4.64 


5.00 


5.68 


6.32 


24 


2.26 


2.49 


2.73 


2.96 


3.41 


4.29 


4.71 


5.12 


5.52 


6.29 


7.01 


27 


2-55 


2.82 


3.09 


3.35 


3.87 


4.38 


4.88 


5.37 


1-^5 


6.31 


7.21 


8.06 


30 


2.85 


3.»S 


346 


3.75 


4-33 


4.91 


5-47 


6.03 


6.57 


7.10 


8.13 


9.12 


33 


3.14 


3.48 


3.82 


4.14 


4.79 


5-44 


6.06 


6.68 


7.29 


7.89 


9.05 


10.2 


36 


3-44 


3.81 


4.18 


4.54 


5.25 


5.96 


6.66 


7.34 


8.01 


8.68 


9.97 


II. 2 


39 


3-74 


4.14 


4.54 


4.93 


5.72 


6.49 


7.2s 


8.00 


8.74 


9.47 


10.9 


12.3 


42 


4-03 


4.47 


4.90 


5-33 


6.18 


7.01 


7.84 


8.66 


9.46 


10.3 


II. 8 


13.3 


45 


4.33 


4-79 


5.26 


5-72 


6.64 


Z-5* 


8.43 


9.31 


10.2 


II. I 


12.7 


14.4 


48 


4.62 


5.12 


5.62 


6.12 


7.10 


8.07 


9.02 


9.98 


10.9 


II. 8 


13.7 


15.4 


51 


4.92 


5-^5 


5.98 


6.51 


7.56 


8.59 


9.61 


10.6 


1 1.6 


12.6 


14.6 


16.5 


54 


5.22 


5.78 


6.35 


6.91 


8.02 


9.12 


10.2 


"3 


J2.4 


134 


15.5 


17.5 


57 


5-51 


6.11 


6.71 


7.30 


8.48 


9.64 


10.8 


11.9 


^K 


14.2 i 16.4 


18.6 


60 


5.81 


6.44 


7.07 


7.70 


8.94 


10.2 


11.4 


12.6 


13.8 


15.0 173 


19.6 



CAST-IRON CYLINDERS. 



257 



Table No. 86 {continued). 
By External Diameter, i Foot Long. 



Ext. 

DiAM. 



i%. in. 

59 
60 

63 
66 

69 
70 

76 
80 
86 
90 

96 
100 
106 

11 o 

116 

12 o 
130 
140 

150 

160 
17 o 

180 

190 
200 



THICKNESS IN Inches. 



^% 



cwts. 

6.10 
6.40 
6.70 
7.00 

7.29 

7.58 

7.88 
8.17 

8.77 

9.36 

9-95 
10.5 

II. I 
11.7 
12.3 
12.9 

13-5 
14. 1 

15.3 
16.5 

17-7 
18.8 

20.0 

21.2 

22.4 
23.6 



iX 



cwts. 

6.77 
7.09 
7.42 

7-75 

8.08 
8.41 

8.74 
9.07 

9.72 
10.4 

II. o 

11.7 

12.3 
13.0 
13-7 
14-3 

15.0 
15.6 
16.9 
18.3 

19.6 
20.9 
22.2 

23.5 

24.8 
26.1 



^H 



CWIS. 

7.43 

7-79 
8.15 

8.51 

8.88 
9.24 
9.60 
9.96 

10.7 
1 1.4 
12. 1 
12.9 

13.6 

14.3 
150 

15.7 

16.5 
17.2 
18.6 
20.1 

21.5 
23.0 
24.4 

25.9 

27.3 
28.8 



iK 



cwts. 

8.09 
8.48 

8.88 
9.27 

9.67 
0.1 
0.5 
0.9 

1.6 
2.4 

3-2 

4.0 
4.8 

5.6 

6.4 

7.2 

7-9 
8.7 
20.3 
21.9 

235 
25.0 

26.6 

28.2 

29.8 
31-4 



i^ 



cwts. 
9.40 
9.86 
0.3 
0.8 

1.2 

1.7 
2.2 

2.6 

35 
4.5 
5-4 
6.3 

7.2 

8.1 

9.1 
20.0 

20.9 
21.8 

237 
25.5 

27.3 
29.2 

31.0 
32.9 

34.7 
36.5 



2 


^% 


cwts. 


cwts. 


10.7 


12.0 


II. 2 


12.6 


II. 8 


13.2 


".3 


13-8 


12.8 


•14.4 


13.3 


14.9 


13.9 


15.5 


;t^ 


16. 1 


154 


^*1W 


16.5 


J^S 


17.5 


19.7 


18.6 


20.8 


19.6 


22.0 


20.7 


23.2 


21.7 


24.4 


22.8 


2S.6 


23.8 


26.7 


24.9 


27.9 


27.0 


30.3 


29.1 


32.7 


31.2 


35.0 


33.3 


?H 


35.4 


;39.8 


37.5 


42.2 


39-6 


44.5 


41.7 


46.9 



^}i 



cwts. 

13-3 
13.9 

14.6 

15.2 

159 
16.6 

17.2 

17.9 



21.5 
23.1 

24.4 
25.7 
27.1 
28.4 

29.7 
31.0 

33-6 
36.3 

38.9 

41.5 
44.2 

46.8 

49-4 
52.0 



»ii 



cwts. 

14.5 

15.2 

15-9 



cwts. 

15.8 
16.6 



cwts. 

18.3 



cwts. 



17 



2^^S 



WEIGHT OF METALS. 



Table No. 87. — ^Volume and Weight of Cast-Iron Balls. 

Given the Diameter. 



Diameter. 


Contents. 


Weight. 


Diameter. 


Contents. 


Weight. 


Diameter. 


Contents. Weight. 


inches. 


cubic 
inches. 


pounds. 


inches. 


cubic 
inches. 


pounds. 


inches. 


cubic feet. 


cwts. 


I 


.524 


.136 


8 


268.1 


69.8 


19 


2.078 


8.35 


1/3 


1.77 


.460 


sy2 


321.5 


•83.7 


20 


2.424 


9-74 


2 


4.19 


1.09 


9 


381.7 


99.4 


21 


2.806 


11.28 


2y2 


8.18 


2.13 


9}4 


448.9 


I16.9 


22 


3.227 


12.97 


3 
3}^ 


14. 1 


3.68 
5-85 


10 


523.6 


136.4 


23 


3,688 
4.188 


14.82 

y 


22.5 


inches. 


cubic feet 


cwts. 


24 


16.83 


4 


33.5 


8.73 


II 


•403 


1.62 


25 


4-736 


19.03 


4>^ 


47-7 


12.4 


12 


.524 


2.10 


26 


5-327 


21.40 


5 ^ 


65-5 


17.0 


13 


.666 


2.68 


27 


5-963 


23.96 


sH 


87.1 


22.7 


14 


.832 


3-34 


28 


6.651 


26.72 


6 


113-1 


295 


15 


1.023 


4. 1 1 


29 


7.390 


29.69 


6}^ 


143-8 


37.5 


16 


1. 241 


4.99 


30 


8.181 


32.87 


7 


179.6 


46.8 


17 


1.489 


5-98 


31 


9.027 


36.27 


1/2 


220.9 


57.5 


18 


1.767 


7.10 


32 


9930 


39-90 



Nbie. — To find the weight of balls of other metals, multiply the weight given in the 
table by the following multipliers : — 

For Wrought Iron 1.067, making about 7 per cent. more. 

Steel 1.088 ,, 9 „ 

Brass 1.12 ,, 12 ,, 

Gun Metal 1. 165 „ 16)^ „ 



Table No. 88. — Diameter of Cast-Iron Balls. 

Given the Weight. 



.Weight 


Diameter. 


Weight 


Diameter. 


Weight 


Diameter. 


Weight 


Diameter. 


pounds. 


inches. 


pounds. 


inches. 


• 

pounds. 


inches. 


cwis. 


inches. 


>^ 


1-54 


14 


4.68 


80 


8.37 


8 


18.73 


I 


1-94 


16 


4.89 


90 


8.71 


9 


19.48 


2 
3 

4 
5 


2.45 
2.80 

3.08 
332 


18 
20 

25 
28 


5-09 

5-27 

5.68 
5-90 


100 


9.02 


10 

12 

14 
16 


20.17 
21.44 
22.57 
23.60 


cwts. 

I 


inches. 

9-37 
10.72 


6 


3.53 


30 


6.04 


2 


11.80 


18 


24.54 


7 


372 


40 


6.64 


3 


13-51 


20 


25.42 


8 


3-89 


50 


7.16 


4 


14.87 


25 


27.38 


9 


4.04 


56 


7-43 


5 


16.02 


30 


29.10 


10 


4.19 


60 


7.60 


6 


17.02 


35 


30.64 


12 


4.45 


70 


8.01 


7 


17.91 


40 


32.03 



WEIGHT OF FLAT BAR STEEL. 



259 



Table No. 89. — Weight of Flat Bar Steel. 

I Foot Long. 













Width in Inches. 










Thickness. 






















>i 


H 


• 


H 


I 


iX 


i>^ 


iH 




mches. 


lb. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 




X 


.425 


.533 


.640 


.743 


.850 


1.06 


1.28 


1.49 




s/i< 


•53i 


.665 


.800 


.929 


1.06 


1.33 


1-59 


1.86 




^ 


.638 


.798 


.960 


1. 1 1 


1.28 


"•59 


1.91 


2.23 




7/16 


.744 


.931 


I.I2 


1.30 


1.49 


1.86 


2.23 


2.60 




}i, 


.850 


1.06 


• 1.28 


1.49 


1.70 


2.13 


^•55 


2.98 




9/16 




1.20 


1.44 


1.67 


I.91 


2.39 


2.87 


3-35 




><, 


^_ 


1-33 • 


1.60 


1.86 


2.12 


2.66 


3- 19 


3-72 




T 






1.76 


2.04 


2.34 


2.92 


3-5* 


4.09 




— 




1:92 


2.23 


2.5s 


3.19 


3.83 


4.46 




t' 


— 


— 


— 


2.41 


2.76 


3-45 


4.14 


4.83 




— 






2.60 


2.98 


372 


4.46 


5-^1 




»5/i6 






— 


— 


319 


3.98 


4.78 


5.58 




I 


— — 




— 


— 


3-40 


4^25 


5.10 


5-95 












Width ii 


4 Inches. 










Thickmkss. 




















Jb ^^ *^»4^ ** Awa^k^v 


2 


2X 


2)i 


2H 


3 


3X 


3>i 


4 




inches. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 




H, 


1.70 


I.9I 


^•33 


2.34 


2.55 


2.76 


2.98 


340 




s/i6 


2.13 


2.39 
2.87 


2.66 


2.92 


3- 19 


3.45 


372 


4.25 




H 


2.98 


3.19 


3.51 


3.83 


4.14 


4.46 


5.10 




1/16 


3.35 


372 


4.09 


4.46 


4.83 


5.21 


^P 




yi 


3.40 


3-83 


4.25 


4.68 


5.10 


5-53 


5-95 


6.80 




9/16 


3.83 


4-30 


4.78 


5.26 


5.74 


6.22 


6.69 


7.6s 




H 


4.25 


4.78 


5-3' 


5.84 


6.38 


6.91 


7.44 


8.50 




T 


4.68 


5.26 


5.84 


6.43 


7.01 


7.60 


8.18 


9.35 




5.10 


5.74 


6.38 


7.01 


7.65 


8.20 


8.93 


10.2 




«3/i« 


5.53 


6.22 


6.91 


7.60 


8.29 


8.98 


9.67 


II. I 




n 


5.95 


6.69 


7-44 


8.18 


8.93 


9.67 


10.4 


II.9 




r^'^ \ It 


7.17 


7-97 


8.77 


9.56 


10.4 


II. 2 


12.8 




7.65 


8.50 


9.35 


10.2 


II. I 


11.9 


13-6 




^P** *>««^ ^v ^ tf <* 






Width n 


r Inches. 










4>i 


5 


S}i 


6 


eyi 


7 


7K 


8 




iiicii<es. 1 Ite- 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 




5^ 1 3.82 


4.26 


4.68 


§•'2 


5.52 


5.96 


6.38 


6.80 


V.<s 4.78 


§•32 


5.84 


6.38 


t'^ 


7-44 


7.97 


8.50 


H 5.74 


6.38 


7.02 


7.66 
8.92 


8.28 


8.92 


9.56 


10.2 


7/16 6.70 


7.44 


8.18 


9.66 


10.4 


II. 2 


II.9 


1 5^ 1 7'^ 


8.50 


9.36 


10.2 


II. I 


II. 9 


12.8 


136 




9/z6 8.60 


9.56 


10.5 


II.5 


12.4 


13.4 


143 


iS-3 




^ 9.56 


10.6 


n.7 


12.8 


13.8 


14.9 


15-9 


17.0 




i«/i6 


10.5 


11.7 


12.9 


14.0 


^H 


16.4 


17.5 


18.7 


yr 1 


II.5 


12.8 


14.0 


15-3 


16.6 


17.9 


19. 1 


20.4 




>3/i6 


12.4 


13.8 


15.2 


16.6 


18.0 


19-3 


20.7 


22.2 




■*■/■ «-^ 

^ 


13-4 


14.9 


16.4 


17.9 


19.4 


20.8 


22.3 


23.8 




«5/x6 


14.3 


15.9 


'2-5 


19. 1 


20.8 


22.4 


23.9 


25.6 




I 


15.3 


17.0 


18.7 


20.4 


22.1 


23.8 


25.5 


27.2 



26o 



WEIGHT OF METALS. 



Table No. 90. — ^Weight of Square Steel. 
I Foot in Length. 



Size. 



inches. 

V16 



/A- 
V.6 

H 

V.6 



'V.6 



Weight. 



pounds. 

•053 
.119 

.212 

•333 
.478 
.651 

.850 

1.08 

1.33 
1. 61 

1.92 

2.24 

2.60 



Size. 



inches. 



'A6 

>^ 

v.6 
I/' 



/'4 

H 

v.6 



2 



Weight, 



pounds. 
3.06 

3-40 

3.83 

4.30 

4.79 

5.31 
5.86 

6.43 
7-03 
7.65 
8.30 
8.98 

9-79 



Size. 



inches. 

2 

2}i 

2^ 

2^ 
2^ 

3 



Weight. 



pounds. 
10.4 
II. 2 
II. 9 
12.8 
13.6 

15-4 
17.2 

19.2 
21.2 

23-5 

25-7 
28.2 

30.6 



Size. 



inches. 



4 
% 

4K 



4' 



6 



Weight. 



pounds. 

35-9 
41.6 

47.8 

54-4 
61.4 

68.9 

76.7 

85.0 

93-7 
102.8 

112.4 

122.4 



Table No. 91. — Weight of Round Steel. 
I Foot in Length. 



Diameter. 


Weight. 


Diameter. 


Weight. 


Diameter. 


Weight. 


Diameter. 


Weight, 


inches. 


pounds. 


inches. 


pounds. 


inches. 


pounds. 


inches. 


cwts. 


'A 


.0417 


I V.6 


5.18 


4 


42.7 


12 


3.433 


V.6 


.0939 


^H 


6.01 


^y^ 


48.3 


I2>^ 


3729 


H 


.167 


I V.6 


6,52 


A% 


54-6 


13 


4.029 


V.6 


.260 


Ifi 


7^o5 


4H 


60.3 


is'A 


4.345 


H 


•375 


I 'V.6 


7.61 


5 


66.8 


14 


4.682 


V.6 


•511 


I^ 


8.18 


1% 


73^6 


I4j4 


5-013 


y2 


.667 


I 'V.6 


8.77 


sy 


80.8 


IS 


. 5-364 


v.6 


.845 


I^ 


9.38 


SVa 


88.3 


^sH 


5-728 


'V.6 


1.04 


I 'V.6 


lO.O 


6 


96.1 


• 16 

i6>^ 
17 


6.103 
6.471 

6.868 
7.302 


1.27 

^•50 
1.76 


2 


10.7 
12.0 

13.6 


inches. 

eyi 
7 

8 


cwts. 

1.007 
I 168 


'V.6 


2.04 
2.35 


2^ 


16.7 


I.34I 
1.526 


18 
19 


7-724 
8.607 


I 


2.67 


2fi 


18.4 


sy 


^•723 


20 


9-537 


I V16 


3.00 


2J^ 


20.2 


9 , 


^•931 


21 


10.52 


i}i 


3^38 


2^1 


22.0 


9y 


2.152 


22 


11-54 


I VI6 


3^76 




24.1 


10 


2.385 


23 


12.62 


I^ 


4.17 


3^ 


28.3 


loj^ 


2.629 


24 


13-73 


I V16 


4.60 


3>^ 


32.7 


II 


2.884 






I?^ 


5-05 


3?< 


34^2 


ii>^ 


3^i5o 







WEIGHT OF CHISEL STEEL. 



261 



Table No. 92. — Weight of Chisel Steel — Hexagonal, Octagonal, 

AND Oval-Flat. 

I Foot in Length. 



Dxaneter 

across the 

Sides. 


HbX AGONAL SbCTION, 




Octagonal Section. 


Sectional Area. 


Weight. 


Length 

to weigh 

I cwt. 


Sectional Area. 


Weight. 


Leng^th 

to weigh 

I cwt 


inches. 


square inches. 


pounds. 


feet 


square inches. 


pounds. 


feet 


H 


.1217 
.2165 

.3383 
.4871 

.6631 


.414 
.736 
I.I5 

1.66 
2.25 


245 

138 
88.3 

61.3 

45 


.1164 

.2070 

.3236 

.4659 
.6342 


.396 
.704 

I.IO 

1.58 

2.16 


268 

151 

96.5 

67 

49.3 


''H 


. .8661 
1.096 


2.94 
3-73 


34.5 
27.3 


.8284 
1.048 


2.82 
3.56 


37-7 
30 




1.353 
1.637 

1.949 


4.60 

5-57 
6.63 


22.5 
18.3 
153 


1.294 

1.566 

1 1.864 


4.40 

5.32 
6.34 


24 

20 

16.8 


t 


Oval-Flat Section. 








inch. inch. 








I x)^ 


.2510 

.4463 
.6974 


.853 
1.52 
2.37 


119 
67 
43 









Table No. 93. — Weight of one Square Foot of Sheet Copper. 

To Wire-Gauge employed by Williams, Foster, & Co. 
Specific Weight taken as 1. 16 (Specific Weight of Wrought Iron = i). 



Thickness. 


Weight of! 
X Square i 
Foot 


Thickness. 


Weight of 

I Square 

Foot. 


Thickness. 


Weight of 

X Square 

Foot 


Wire- 
Gauge. 

No. 


Inch 
(approxi- 
mate). 


pounds. 


Wire- 
Gauge. 

No. 


Inch 
(approxi- 
mate). 


pounds. 


Wire- 
Gauge. 

No. 


Inch ^ 

(approxi- 
mate). 


pounds. 


I 


.306 


14.0 


II 


.123 


565 


21 


.0338 


1-55 


2 


.284 


13.0 


12 


.109 


5.00 


22 


.0295 


1-35 


3 


.262 


12.0 


13 


.0983 


4.50 


23 


.0251 


i-iS 


4 


.240 


II. 


14 


.0882 


4.00 


24 


.0218 


I.OO 


5 


.222 


10.15 


15 


.0764 


3.50 


25 


.0194 


.89 


6 
7 


.203 
.186 


930 
8.50 


16 
17 


.0655 
.0568 


3.00 
2.60 


26 
27 


.0172 
.0153 


•79 

.70 


8 


.168 


7.70 


18 


.0491 


2.25 


28 


.0135 


.62 


9 


•153 


7.00 


19 


.0437 


2.00 


29 


.0122 


•56 


10 


.138 


6.30 


20 


.0382 


1.75 


30 


.0110 


•50 



i WEIGHT OF METALS, 

Table No. 94. — Weight of Copper Pipes and Cylinders, 

BY Internal Diameter. 

Length, i Foot. Thickness by HoIuapHera Wite-Gauge (Table No. 13). 

Spedfic Weight-i.l6 (Specific Wdght of Wrought-Iron-i). 



Di^H. 


Ito. 


Ibt 


ibi. 


Ila. 


Ibi. 


Ita. 


lbs. 


Lbs. 


lbs. 


ibt. 
























H 


3- "4 


2.84 


2-33 


1.92 


1.53 


I.4I 


1.21 


1.05 


■934 


f?i 


% 


3-84 


3-49 


2.91 


2.44 


1,99 


1.84 


1.60 


1.41 


1.27 




H 


4-54 


4.13 


3-49 


2-95 


2-45 


2.27 




1.77 


1.60 


1-43 


'A 


5-^3 


4.78 


4.06 


3-47 


2.91 


2.71 


2-39 


2.13 


•93 


1-73 


H 


5-93 


5.42 

6.07 


4.64 


3-99 


3-37 
3-83 


3- '4 


2.7E 


It 


2.26 


2.04 




6.63 


5-22 


4.50 


3-57 


3. "7 


2.60 


2.3s 




7.32 


6.71 


5-79 


S.02 


4-29 


4.00 


3.57 


3.22 


2.93 


2.66 


' 




7.36 


6.37 


5-53 


4.74 


4.43 


3-96 


3-57 


3-26 


2.97 


iH 


8.71 


8.00 


6.95 


6.05 


S.20 


4. 86 


435 


394 


3.60 


3.28 


"X 


9.4D 


8,6s 


7.52 


6.57 


ii\ 


S-29 


4-75 


4-30 


m 


]:g 


'H 




9.30 


8.10 


7.08 


S-72 


5-14 


4.66 


t'A 


lo's 


9-94 


8.68 


7,60 


6.57 


6,16 


5-53 


S-02 


4.60 


4.20 


'H 


ii.S 


10.6 


9.26 


8.12 


7.02 


6-59 


5-93 


5-39 


4-93 


til 


iH 






9.83 


8.63 


7.48 


7.02 


6.32 


5-75 


S-27 


m 


ia!s 


11.9 


10,4 


9-15 


7.93 


;:si 


6.71 


6.11 


5.60 


S.I2 




'3.5 


12.S 




9.66 


8-39 


7.11 


6.47 


5-93 


5-43 


2H 


14.2 


i3.a 


11.6 


10.2 


8.84 


8.31 




6.83 


6.27 


5-74 


i% 


14.9 


13-8 




10.7 


9-3° 


8.74 


7- "9 


6.60 


6.05 


2H 


'5-6 


14-5 


12,7 




9-75 


9- '7 


7-56 


6.94 


6.36 


'H 


16.3 


IS- 1 


13.3 


niy 




9.60 


7.92 


7.27 


6.67 


'H 


17.0 


158 


•3-9 


122 


10.7 


10.0 


9.07 


8. 38 


7.60 


',% 


^H 


17.7 


16.4 


•4-5 


ii^S 




10.5 


9-47 


8.64 


7.94 


aji 


.§.4 


17. 1 


iS-o 


13-3 
13-8 


II-S7 


10.9 


9.86 


9.00 


8.27 


7- 59 


3 


"9-1 


17-7 


15.5 




"■3 


10.3 


9.36 


8.61 


7.90 


3J4 


iO.4 


19.0 


16.8 


148 


12.9 


ii,2 


U.I 


10.1 


9.17 


8.52 


3H 


21.8 




17-9 


•5-9 


51 


131 


11.8 


10.8 


9-94 


9-13 


3K 


23-2 


21.6 


19. 1 


16.9 


m 


12.6 


ii-s 


ia.6 


9-75 


4 


24.6 


22.9 




17.9 


'5-7 


•3-4 


12.3 


"-3 


10.4 


4X 


15.9 


24.2 


21.4 


19,0 


16.6 


:« 


14-Z 


13-0 


12.0 


110 


A'A 


It', 


r, 


22.5 




'^5 


'5-2 


'3-7 


12.7 


11.6 


^M 


23-7- 
24-8 




18.4 


17.4 


IS.8 


14.4 


13-3 




5 


30-1 


28,0 


22.1 


19.3 


18.Z 


16.6 


IS-I 


14.0 


12.8 


%%, 


32.8 


29-3 


26.0 


23-1 


10.2 


19. 1 


17-3 


15-9 


14.6 


'3-5 


s'A 


30.6 


27.1 


24.1 






18. t 


16.6 


«S-3 


141 


SU 


34.2 


31-9 


28.3 


25.2 




2a!8 


.8.9 


;i:5 


16.0 


14.7 


6 


356 


33-2 


29.5 


26.2 


23.0 


'■•' 


19-7 


16.6 


'5-3 



S.04, 

5-32 



WEIGHT OF COPPER PIPES AND CYLINDERS. 



263 



Table No 94 {contintied). 

Length, i Foot. Thickness by Holtzapffers Wire- Gauge (Table No. 13). 
Specific Weight = 1. 16 (Specific Weight of Wrought Iron= i). 



Thick- 




























kiss. 


8 


9 


10 


II 


la 


13 


14 


15 


16 


17 


18 


19 


ao 


W.G. 




























Inch. 


.165 


.148 


•134 


.120 


.109 


.095 


.083 


.072 


.065 


.058 


.049 


.042 


•035 


"/64* 


9/6</ 
lbs. 


9/64^ 


H^ 


7/64 


3/3a/ 


lbs. 


5/64^ 

lbs. 


'A6/ 


Vi6^ 


3/64/ 


3/64^ 


^M 


IST. 
DlAM. 

,bches. 


Ihs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


.581 


.491 


.422 


•357 


.310 


.254 


.210 


.173 


.150 


.129 


.104 


.086 


.068 


X 


.832 


.716 


.626 


.540 


.476 


.398 


.336 


.282 


.249 


.217 


.178 


.149 


.121 


H 


1.08 


.941 


.830 


.722 


.641 


•553 


.462 


•391 


.348 


•305 


.253 


.213 


!228 


a 


1-33 


I.I7 


1.03 


.904 


.807 


.687 


.588 


.500 


.447 


.393 


.327 


•277 


H 


1.58 


1.39 


1.24 


1.09 


.972 


.831 


•Z'4 


.610 


.545 


.481 


.402 


.341 


.281 


)i. 


1-83 


1.62 


1.44 


1.27 


1. 14 


.975 


.840 


.719 


.644 


.570 


.476 


.404 


•334 


}i 


2.09 


1.84 


1.65 


1.45 


1.30 


1. 12 


.966 


.828 


•743 


.658 


•550 


.468 


•387 


I 


2.34 


2.05 


1.85 


1.63 


1.47 


1.26 


1.09 


.938 


.842 


.746 


.625 


.532 


.440 


iH 


^:i^ 


2.27 


2.05 


1.82 


1.63 


1. 41 


1.22 


1.05 


.940 


.834 


.699 


.596 


•493 


iH 


2.49 


2.26 


2.00 


1.80 


1-55 


1.34 


1. 16 


1.04 


.922 


•774 


.659 


•547 


'^, 


309 


2.72 


2.46 


2,18 


1.97 


1.70 


1.47 


1.27 


1. 14 


I.OI 


.848 


.723 


.600 


'K 


3.34 


2.94 


2.67 


2,36 


2.13 


1.84 


1.60 


1.38 


1.24 


1. 10 


.922 


•787 


.653 


•K 


359 


3-17 


2.87 


2.55 


2.30 


1.99 


1.72 


1.48 


1-34 


1. 19 


•997 


.851 


.706 


 % 


3.&4 


3-39 


3-07 


2.73 


2.46 


2.13 


1.85 


1.59 


1.43 


1.27 


1.07 


•9^5 


•759 


^a 


4.09 


3.62 


3.28 


2.91 


2.63 


2.27 


1-97 


1.70 


^53 


1.36 


1.15 


.978 


.812 


2 


434 


3.84 


3.48 


309 


2.79 


2.42 


2.10 


1. 81 


1.63 


1.45 


1.22 


1.04 


.865 


t% 


459 


407 


3.69 


327 


2.96 


2.56 


2.23 


1.92 


H^ 


1.54 


1.29 


i.ii 


.919 


i'4 


484 


429 


3.89 


3-45 


3- 12 


2.71 ; 


^•35 


2.03 


1.83 


1.63 


1.38 


1.17 


.972 


% 


5-09 


4.52 


4.09 


3-64 


329 


2.85 


2.48 


2.14 


1-93 


H' 


1-45 


1-23 


1.03 
1.08 


^}i 


5-34 4.74 


4.30 


3.82 


3-45 


3.00 


2.60 


2.25 


2.03 


1.80 


1.53 


1.30 


»H 


l-s 


497 


4.50 


4.00 


3-62 


3.14 


2.73 


2.36 


2.13 


1.89 


1.60 


1.36 


1. 13 


^H 


519 


4.71 


418 


3-79 


3.28 


2.86 


2.47 


2.22 


1.98 


1.68 


1-43 


1. 18 


m 


6.09 


5.42 


4.91 


4.37 


3-95 


3-43 


2.98 


2.58 


2.32 


2.07 


H^ 


1.49 


1.24 


3 


6-34 


5.66 


5.11 


4-55 


4.12 


3.57 


3" 


2.69 

1 


2.42 


2.16 


1.82 


1-55 


1.29 


^ 


6.8s 


6.11 


5.52 


4.91 


4-45 


3.86 


3.36 


2.91 


2.62 


2.33 


1.96 


1.68 


1.40 


3^ 


7.8s 


6.56 


5-93 


5.28 


4.78 


4.15 


3.61 


3.12 


2.82 


^•5J 


2. 1 1 


1.81 


1.51 


3¥ 


7.01 


6.33 


5.64 


5.11 


4.44 


3.87 


3.34 


3.01 


2.68 


2.26 


1.94 


1.62 


4 


8-35 


7.46 


6.74 


O.OI 


5.44 


4.73 


4.12 


3.56 


3-21 


2.86 


2.41 


2.06 


173 


4)( 


8.85 


7-9» 


7.14 


6.37 


5-77 


5.02 


4.37 


3.78 


341 


3-<H 


2.56 


2.19 


1.84 


*^/ 


9-35 


8.36 
8.81 


7-55 


6.74 


6.10 


5-30 


4.62 


400 


3.61 


3.21 


HI 


2.32 


1.94 


aH 


9.85 


7.96 


7.10 


6.43 


5-59 


487 


4.22 


3.80 


3-39 


2.86 


2.45 


2.05 


5 


10.4 


9.26 


8.36 


7.46 


6.77 


5.88 


5.^3 


4.44 


4.00 


356 


301 


2.57 


2.16 


5^ 


10.9 


9.71 


8.77 


7.83 


7.10 


6.17 


5.38 


4.66 


420 


3-74 


3. IS 


2.70 


2.27 


5^ 


11.4 


ia2 


9.18 


8.19 


7.43 


6.46 


I'M 


4.88 


4.40 


392 


3.30 


2.83 


2.38 


I^ 


11.9 


10.6 


9.58 


8.56 


7.76 


6.7s 


5.09 


4.59 


4.09 


3.45 


2.96 


2.48 


6 


•M 


II. I 


9-99 


8.92 


8.09 


7.04 


6.14 


5.31 


4.79 


4.27 


3-6o 


3-09 


2.58 



262 



WEIGHT OF METALS. 



Table No. 94. — ^Weight of Copper Pipes and Cylinders, 

BY Internal Diameter. 

Length, i Foot. Thickness by Holtzapffel's Wire-Gauge (Table No. 13). 
Specific Weight = 1. 16 (Specific Weight of Wrought-Iron= i). 



Thick- 

KESS. 

W. G. 



Inch. 



Int. 

DlAM. 

inches. 

H 

H 

n 



1% 

i}i 
iH 

2 

2)i 
2H 
2H 
2>^ 

2H 
2H 
2H 

3 

3X 
3'A 
3H 
4 

A'A 

aH 

5 

5^ 
SA 

sH 

6 



0000 



•454 

39/64 



lbs. 

3H 
3-84 
4-54 
5-23 

5-93 
6.63 

7.32 
8.02 

8.71 
9.40 
10. 1 
10.8 

11.5 
12. 1 
12.8 

>3.S 

14.2 
14.9 
15.6 
16.3 

17.0 
17.7 
18.4 
19. 1 

20.4 
21.8 
23.2 
24.6 

25.9 

273 
28.7 

30.1 

3'-S 
32.8 

34.2 
35-6 



000 


00 


.425 

'7/64/ 


.380 


lbs. 


lbs. 


2.84 
3-49 

4.78 


2.33 

2.91 

3-49 
4.06 


5.42 
6.07 
6.71 
7.36 


4.64 
5.22 

5.79 
6.37 


8.00 
8.65 

9.30 
9.94 


6.95 

7-52 
8.10 

8.68 


10.6 
II. 2 
11.9 
12.S 


9.26 

9.83 
10.4 

II.O 


13.8 

H-5 
15.1 


1 1.6 
12. 1 
12.7 

13.3 


15.8 
16.4 
17.1 
17.7 


13.9 

'4.5 
15.0 

156 


19.0 
20.3 
21.6 
22.9 


16.8 
17.9 
19. 1 
20.2 


24.2 

25.4 
26.7 

28.0 


21.4 
22.5 

23.7- 
24.8 


29.3 
30.6 

31-9 
33-2 


26.0 
27.1 
28.3 
29.5 



.340 

11/32 



lbs. 

1.92 

2.44 
2.95 

3-47 

3-99 
4.50 

5.02 
5-53 

6.05 
6.57 
7.08 
7.60 

8.12 
8.63 

9.15 
9.66 

10.2 
10.7 
II. 2 
11.7 

12.2 
12.8 

13.3 
13-8 

14.8 

15.9 
16.9 

17.9 

19.0 
20.0 
21.0 
22.1 

23.1 
24.1 
25.2 
26.2 



.300 

'9/64/ 



lbs. 

1.53 
1.99 

2.45 

2.91 

3.37 

3.83 
4.29 

4.74 

5.20 
5.65 
6. 1 1 
6.57 

7.02 
7.48 
7.93 
8.39 

8.84 

930 

9-75 
10.2 

10.7 
II. I 
11.57 
12.0 

12.9 

139 
14.8 

15.7 
16.6 

18.4 
19.3 

20.2 
21. 1 
22.1 
23.0 



.284 

9/32/ 



lbs. 

1. 41 
1.84 
2.27 
2.71 

3.14 

3-57 
4.00 

4-43 

4.86 
5.29 

572 
6.16 

6.59 
7.02 

7-4S 
7.8g 

8.31 
8.74 

9.17 
9.60 

lO.O 

10.5 
10.9 

"•3 

12.2 
131 

'3-2 
14.8 

15.6 
16.5 

17.4 
18.2 

19. 1 

20.0 
20.8 
21.7 



.259 

X/ 



lbs. 

1. 21 
1.60 
2.00 

2.39 

2.78 
317 

3-57 
396 

4-35 

4.75 
5.14 

5-53 

5-93 
6.32 

6.71 

7.II 

7.50 
7.89 
8.29 
8.68 

9.07 

9.47 
9.86 

10.3 

1 1. 1 
11.8 
12.6 

»3-4 

14.2 
15.0 
15.8 
16.6 

17.3 
18. 1 

18.9 

19.7 



.238 

'5/64/ 



lbs. 

1.05 
I.4I 
1.77 
2.13 

2.50 
2.86 

3-22 
3.57 

3-94 

4-30 
4.66 

5.02 

5.39 

5.75 
6. 1 1 

6.47 

6.83 
7.19 

756 
7.92 

8.28 
8.64 
9.00 
936 

10. 1 
10.8 
II. 5 

12.3 

13.0 

137 
14,4 

15. 1 

15-9 
16.6 

'Z-3 
18.0 



.220 

7/3"/ 



lbs. 

.934 
1.27 
1.60 

1.93 

2.26 
2.60 

2.93 

3-26 

3.60 

3-93 
4.26 

4.60 

4.93 
5.27 

5.60 
5-93 

6.27 
6.60 
6.94 
7.27 

7.60 

7.94 
8.27 

8.61 

9.27 

9-94 
10.6 

"•3 

12.0 
12.7 

13.3 
14.0 

14.6 

15-3 
16.0 

16.6 



.203 

'3/64 






lbs. 

.809 
1. 12 

L43 
1.73 

2.04 

2.35 
2.66 

2.97 

3.28 

3.58 

3.89 
4.20 

4-Si 
4.82 

5.12 
5-43 

5.74 
6.05 

6.36 

6.67 

6.97 
7.28 

7.59 
7.90 

8.52 

9- 13 

9.75 
10.4 

II.O 

II. 6 
12.2 
12.8 

13.5 
14. 1 

14.7 
15.3 



.180 

3/16 b. 



lbs. 

.667 
.941 
1. 21 
1.49 

1.76 
2.03 
2.31 
2.58 

2.85 

313 

340 
3.68 

3.95 
4.22 

4.50 

4-77 

5.04 
5-32 

5.86 

6.14 
6.41 
6.68 
6.95 

7.50 
8.04 

8.59 
9.»3 

9.67 
10.2 
10.8 

"3 

11.9 
12.4 
12.9 

13-5 



WEIGHT OF COPPER PIPES AND CYLINDERS. 



263 



Table No 94 {continual). 

Length, i Foot. Thickness by Holtzapffers Wire- Gauge (Table No. 13). 
Specific Weight = 1. 16 (Specific Weight of Wrought Iron= i). 



Thick- 

KBSS. 

W. G. 

• 


8 


9 


10 


IZ 


12 


13 


14 


15 

.072 

5/64^ 

lbs. 


z6 
.065 

Vi6y 


17 


z8 


Z9 


20 


Inch. 


.165 
lbs. 


.148 

9/64/ 

lbs. 


.134 

9/64^ 


.120 


.109 

7/64 


.095 

3/3>/ 

1 

lbs. 


.083 

lbs. 


.058 


.049 
3/64/ 


.042 

3/64^ 


.035 


Int. 

DiAM. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


inches. 




























i 


.581 
.832 
1.08 

1.33 


.491 
.716 
.941 
I.17 


.422 
.626 
.830 
1.03 


•357 

.540 
.722 

.904 


.310 
.476 

.641 
.807 


•254 
.398 

'.687 


.210 

.336 
.462 
.588 


•173 

.282 

•391 
.500 


.150 

.249 

.348 

.447 


.129 
.217 

•305 

•393 


.104 
.178 

.253 
.327 


.086 

.149 
.213 
.277 


.068 

.121 

•'75 
.228 


i 

I 


1.58 

1-83 
2.09 

2.34 


1-39 
1.62 

1.84 

2.05 


1.24 
1.44 
1.65 
1.85 


1.09 
1.27 

1.63 


.972 
1. 14 

1.30 

1.47 


.831 

.975 
1. 12 

1.26 


.714 
.840 

.966 

1.09 


.610 
.719 
.828 

.938 


•545 
.644 

.743 
.842 


.481 
.570 
.658 
.746 


.402 
.476 

.550 
.625 


•341 
.404 
.468 

.532 


.281 

.334 

.387 
.440 




2.59 
2.84 

3-09 
3-34 


2.27 
2.49 
2.72 
2.94 


2.05 
2.26 
2.46 
2.67 


1.82 
2.00 
2.18 
2.36 


1.80 
1.97 
2.13 


I.4I 

1.55 
1.70 

1.84 


1.22 

1-34 

1.47 

1.60 


1.05 
1. 16 
1.27 
1.38 


.940 
1.04 

1.14 
1.24 


•834 
.922 

I.OI 

1. 10 


.699 

.774 
.848 

.922 


.596 
.659 

.723 
.787 


•493 

.547 
.600 

.653 


2 


4.09 
4-34 


317 

3-39 
362 

3.84 


2.87 

3-07 
328 

3.48 


2.55 

2.73 
2.91 

3-09 


2.30 

2.46 
2.63 

2.79 


1.99 

2.13 
2.27 

2.42 


1.72 

1.85 
1.97 

2.10 


1.48 

1.59 
1.70 

I.81 


1-34 

1.43 

1.53 
1.63 


1. 19 
1.27 

1.36 
1.45 


•997 
1.07 

i.iS 

1.22 


.851 

.978 
1.04 


.706 

.759 
.812 

.865 


2H 


4-59 
4.84 

5.09 
5.34 


4.07 
4.29 

4-74 


3-69 
3.89 
4.09 

4.30 


3-27 
3-45 
3-64 
3.82 


2.96 
3.12 

329 
3.45 


2.56 
2.71 
2.85 
3.00 


2.23 

2.35 
2.48 

2.60 


1.92 
2.03 
2.14 
2.25 


'•Z3 
1.83 

1-93 
2.03 


1-54 
1.63 


1.29 
1.38 
1.45 
1-53 


I. II 
1.17 
123 
1.30 


.919 
.972 
1.03 
1.08 


3 


6.09 
6.34 


4.97 

5.19 
5.42 

5.66 


4.50 

4.71 
4.91 

5.II 


4.00 
4.18 

4.37 
4.55 


362 

3-79 

3.95 
4.12 


3- '4 
3.28 

3-43 
3-57 


2.73 

2.86 
2.98 

3." 


2.36 
2.47 
2.58 
2.69 

1 


2.13 
2.22 

2.32 
2.42 


2.07 
2.16 


1.60 
1.68 

1.82 


1.36 
1-43 
1.49 
1.55 


1. 18 
1.24 
1.29 


3U 
3% 
3H 

4 


6.85 

7.85 
8.35 


6.1 1 

6.56 
7.01 
7.46 


5.52 
5.93 
6.33 
6.74 


4.91 
5.28 
5.64 

O.OI 


4.45 
4.78 
5.11 

5.44 


3.86 

415 
4-44 
4.73 


3.36 
3-6i 

3.87 
4.12 


2.91 
312 

3-34 
3-56 


2.62 
2.82 

3.01 
3.21 


2.33 
2.51 

2.68 
2.86 


1.96 
2. 1 1 
2.26 
2.41 


1.68 
1. 81 
1.94 
2.06 


1.40 
1.51 
1.62 

1.73 


4X 

4K 

4H 
5 


8.85 

9-35 
9.85 

10.4 


7.91 
8.36 

8.81 

9.26 


7.14 

7.55 
7.96 

8.36 


6.37 
6.74 

7.10 

7.46 


6.10 

6.43 
6.77 


5.02 
5-30 

5-5§ 

5.88 


4.37 
4.62 

487 
5.13 


3.78 
4.00 
4.22 

4.44 


341 
3.61 

3.80 

4.00 


3.04 
3.21 

3.39 
356 


2.56 
2.71 
2.86 
3.01 


2.19 
2.32 

2.45 
2.57 


1.84 

1.94 
2.05 
2.16 




10.9 
11.4 
11.9 
12.4 


9.71 
ia2 
10.6 
II. I 


8.77 
9.18 

9.58 

9.99 


7.83 

8.19 
8.56 
8.92 


7.10 

7.43 
7.76 

8.09 


6.17 
6.46 

6.75 
7.04 


5.38 

6.14 


466 
4.88 
5.09 
5-31 


4.20 
4.40 

459 
4.79 


3^74 
392 
4.09 
427 


3.15 
330 

3.60 


2.70 
2.83 
2.96 

3-09 


2.27 

2.38 
2.48 
2.58 



264 



WEIGHT OF METALS. 



Table No. 94 {continued). 

Length, i Foot. Thickness by Holtzapffel's Wire-Gauge (Table No. 13). 
Specific Weight=i.i6 (Specific Weight of Wrought- Iron =1). 



Thick- 
























ness. 


0000 


000 


00 





z 


2 


3 


4 


5 


6 


7 


W. G. 


























•454 

»6/64 


.425 


.380 


.340 


.300 


.284 


.259 


.238 


.220 


.203 


.180 


Inch. 


'7/64/ 


^/ 


"/33 


19/64/ 


9/3^/ 


x/ 


'5/64/ 


Vs"/ 


^3/64 


3/16 b. 


Int. 








• 














 


DiAM. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


inches. 
























eVz 


38.4 


35.8 


31.8 


28.3 






21.3 


19.5 


18.0 




14.6 


7 


41. 1 


38.3 


34.1 


30.3 


26.6 


25.1 


22.8 


20.9 


19.3 


17.8 


15-7 


VA 


43-9 


40.9 


36.4 


32.4 


28.4 






22.4 


20.6 




16.8 


8 


46.6 


43-5 


38.7 


34.5 


303 


28.6 


26.0 


23.8 


22.0 


20.2 


17-9 


9 


52.1 


48.7 


43-3 


38.6 


33.9 


32.0 


29.1 


26.7 


24.6 


22.7 


20.1 


10 


57.7 


53.8 


479 


^•2 


37.5 


35-5 


32.3 


29.6 


273 


25.2 


22.2 


11 


^2-^ 


59-0 


52.5 


46.8 


41.2 


38.9 


35-4 


32.5 


30.0 


27.7 


24.4 


12 


68.7 


64.2 


57.2 


51.0 


44.8 


42.4 


38.6 


35.4 


32.7 


30.1 


26.6 


13 


74.2 


69.3 


61.8 


55.1 


48.5 


45.8 


41.7 


38.3 


35.3 


32.6 


28.8 


14 


79.7 


74.5 


66.4 


59-2 


52.1 


49.3 


44.9 


41.2 


38.0 


35-1 


31.0 


15 


85.2 


P't 


71.0 


63.4 


55.8 


52.7 


48.0 


44.1 


40.7 


37.6 


33-2 


16 


90.7 


84.8 


75.6 


67.7 


59^4 


56.2 


51.2 


46.9 


43-4 


40.0 


35-4 


17 


96^3 


90.0 


80.2 


71.8 


63.0 


59.6 


54.3 


49.8 


46.0 


42.5 


37.5 


18 


101.8 


95- 1 


84.9 


76.0 


66.7 


63.1 


57.4 


52.7 


48.7 


45.0 


39.7 


19 


107.3 


100.3 


89.5 


80.1 


70.3 


66.5 


60.6 


5g.6 


51.4 


47.4 


41.9 


20 


112.8 


105.5 


94.1 


84.2 


74.0 


70.0 


63.7 


58.5 


54.0 


49-9 


44.1 


21 


118.3 


1 10. 7 


98.7 


88.3 


77.6 


73.4 


66.9 


61.4 


56.7 


52.4 


46.3 


22 


123.8 


115.8 


103.3 


92.5 


81.3 


76.9 


70.0 


64.3 


59.4 


54.9 


48.5 


23 


129.3 


120.9 


107.9 


96.6 


84.9 


80.3 


73-2 


67.2 


62.1 


57.3 


50.7 


24 


I34^8 


1 26. 1 


112.6 


100.6 


88.6 


83.8 


76.3 


70.1 


64.7 


59.8 


52.9 


26 


146.0 


136.4 


121. 8 


108.8 


95.9 


90.7 


82.6 


759 


70.1 


64.7 


57.2 


28 


^IZ-* 


146.7 


131-0 


117.1 


103. 1 


97.6 


89.0 


81.7 


Z5-4 


69.7 


61.6 


30 


168.4 


157.1 


140.2 


125.4 


1 10.4 


104.5 


95.3 


87.5 


80.8 


74.6 


66.0 


32 


179.6 


167.4 


149.5 


133.6 


117.7 


111.4 


101.6 


93.3 


86.2 


79.6 


70.4 


34 


190.7 


^IV 


158.7 


141.9 


125.0 


1 18.3 


107.9 


99.1 


91.5 


84.5 


74-7 


36 


201.9 


188.0 


167.9 


150. 1 


132.3 


125.2 


114.2 


104.9 


96.9 


89.5 


79.1 



WEIGHT OF COPPER PIPES AND CYLINDERS. 



265 



Table No. 94 {continued). 

Length, i Foot. Thickness by Holtzapffers Wire-Gauge (Table No. 13). 
Specific Weight = 1. 16 (Specific Weight of Wrought Iron= i). 



Thick- 




























mess. 


8 


9 


10 


zz 


Z2 


13 


14 


15 


16 


17 


18 


19 


20 


W. G. 




























Inch. 


.165 


.148 


.134 


.120 


.109 


.095 


.083 


.072 


.065 


.058 


.049 


.042 


•035 


"/fi*^ 


9/64/ 


9/64^ 




7/64 


3/3>y 


sM 


5/64^ 


V.6/ 


V16* 


3/64/ 


3/64* 


V3«/ 


Int. 

DiAM. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


inches. 




























e>% 


13.4 


12.0 


10.8 


9.65 


8.75 


7.61 


6.64 


5.75 


5- '2 


462 


3-90 


3-34 


2.80 


7 


14.4 


12.9 


II. 6 


10.4 


9.42 


8.19 


7.14 


6.19 


5.58 


497 


420 


3.60 


301 


2^ 


15.4 


13.8 


12.47 


II. I 


10. 1 


8.77 


7.65 


6.63 


5.98 


5.33 


449 


3.85 


3-23 


8 


16.4 


147 


13-2 


II. 8 


10.74 


9.34 


8.15 


7.06 


6.37 


5.68 


479 


4.10 


3.43 


9 


18.4 


'f-5 


14.9 


^3-3 


12. 1 


10.5 


9.16 


7.94 


7.16 


6.38 


5.39 


4.61 


3.86 


10 


20.4 


18.2 


16.5 


148 


13.4 


II. 7 


10.2 


8.81 


7.95 


7.08 


5.98 


5.12 


4.28 


II 


22.4 


20.0 


18. 1 


16.2 


147 


12.8 


II. 2 


9.69 


8.74 


7-79 


6.58 


5.63 


470 


12 


24.4 


21.8 


19.8 


17.7 


16.0 


140 


12.2 


10.6 


9.53 


8.49 


7.18 


6.14 


5.13 


13 


26.4 


23.6 


21.4 


19. 1 


'Z-4 


15.1 


132 


11.4 


10.3 


9.20 


7.77 


6.65 


5.55 


14 


28.4 1 25.4 


23.0 


20.6 


18.7 


16.3 


14.2 


12.3 


II. I 


9.90 


In 


7.16 


5.98 


15 


30.4 


27.2 


24.6 


22.1 


20.0 


17.4 


15.2 


13-2 


II.9 


10.6 


8.96 


7.67 


6.40 


16 


32.4 


29.0 


26.3 


23.5 


21.3 


18.6 


16.2 


141 


12.7 


"3 


9.56 


8..18 


6.82 


17 


34-4 


30.8 


27.9 


25.0 


22.7 


19.7 


17.2 


149 


13.5 


12. 1 


10.2 


8.69 


7.27 


18 


36.4 


32.6 


29.5 


26.4 


24.0 


20.9 1 18.2 

1' 


15.8 


14.3 


12.7 


10.7 


9.20 


7.69 


19 


384 


34.4 


31.2 


27.9 


25.3 


22.0 


19.2 


16.7 


I5.I 


13-4 


II-3 


9.71 


8.12 


20 


40.4 


36.2 


32.8 


29-3 


26.6 


23.2 


20.2 


17.6 


159 


14. 1 


11.9 


10.2 


8.54 


21 


42.4 


38.0 


34.4 


30.8 


27.9 


24.3 


21.3 


18.4 


16.6 


14.8 


12.S 


10.7 


8.96 


22 


4*.4 


39.8 


36.0 


32.3 


29-3 


25.5 


22.3 


19.3 


17.4 


15-5 


131 


II. 2 


9.39 


23 


46.4 


41.6 


37.7 


33.7 


30.6 


26.7 


233 


20.2 


18.2 


16.2 


137 


11.8 


9.81 


24 


48.5 


43-4 


39.3 


35.2 


31-9 


27.8 


24.3 


21. 1 


19.0 


16.9 


143 


12.3 


10.2 


26 


52.5 


47.0 


42.6 


38.1 


34.6 


30.1 


26.3 


22.8 


20.6 


18.4 


15.5 


133 


II. I 


28 


56.5 


50.6 


45.8 


41.0 


37.2 


32.4 


28.3 


24.6 


22.2 


19.8 


16.7 


143 


11.9 


30 


60.5 


54.2 


49.1 


43.9 


39.9 


347 


30.3 


26.3 


237 


21.2 


»7.9 


15.3 


12.8 


32 


^^ 


57.8 


52.3 


46.8 


42.5 


37.0 


32.3 


28.1 


25.3 


22.6 


19. 1 


16.3 


13.6 


34 


68.5 


61.4 


55.6 


49.8 


^5-1 


39.4 


344 


29.8 


26.9 


240 


20.3 


17.4 


145 


36 

1 

1 
1 


72.5 


65.0 


58.8 


52.7 


47.8 


41.7 


36.4 


31.6 


28.5 


25.4 


21.5 


18.4 


15.3 



266 



WEIGHT OF METALS. 



Table No. 95. — ^Weight of Brass Tubes, 

BY External Diameter. 

Length, i Foot. Thickness by Holtzapffel's Wire-Gauge (Table No. 13). 

Specific Weight=i.ii (Specific Weight of Wrought Iron=i). 



Thick- 

NBSS. 

W. G. 


15 


z6 


17 


18 


19 


20 


ax 


aa 


23 


24 


25 


vInch. 


.072 

5/64^ 


.065 


.058 
'/16 b. 


.049 

3/64/ 


.042 
3/64 b. 


.035 

V32/ 


.032 

' V32 


.028 
1/32 b. 


.025 

1.6/64 


.022 

1.4/64 


.020 

1.3/64 


DiAM. 

inches. 


lbs. 


lbs. 


lb. 


lb. 


lb. 


lb. 


lb. 


lb. 


lb. 


lb. 


lb. 


3/16 
s/16 


.201 


.187 


.087 
.130 
.172 


.079 
.115 
.150 


.072 
.102 
.132 


.037 
.062 

.088 
."3 


.o3| 

!o8i 
.104 


.031 
.052 
.072 
.092 


.029 

.047 
.065 

.083 


.026 
.042 
.058 
.074 


.024 

•039 

.053 
.068 


7/16 

9/16 


.306 

.358 
.411 


.234 
.281 

.329 
•376 


.214 
.256 
.298 

.340 


.186 
.221 
.257 

.293 


.163 

•»93 
.224 

.254 


.138 
.164 

.189 
•215 


.128 
.151 
.174 
.197 


."3 
•133 
•154 
.174 


.102 
.120 

.138 
.156 


.090 
.106 
.122 
.138 


.082 
.097 
.III 
.126 


^3/16 


•463 

.515 
.567 

.620 


.423 
.470 

.517 
.564 


.382 

•424 
.467 

.509 


.328 
.364 

•399 
.435 


.285 

.346 
.376 


.240 
.265 
.291 
.316 


.221 

•244 
.267 

.290 


.194 
.215 

•235 
•255 


.174 
.192 

.211 

.229 


.154 
.170 
.186 
.202 


.141 

.155 
.170 

.184 


»5/i6 
I 


.672 
.724 

.'S8i 


.611 
.658 
.706 
.801 


•S5I 

.593 

.635 
.719 


.471 
.506 
.542 
.613 


.407 

.437 
.468 

.529 


.342 
•367 
•393 
.443 


.314 

•337 
.360 

.407 


.276 
.296 
.316 

•357 


.247 

.265 

. .283 

.320 


.218 

.234 
.250 

.282 


.199 
.213 

.228 

•257 




.986 
1.09 
1.20 


.896 

.991 
1.09 


.972 


.684 
.827 


.590 
.651 
.712 


.494 

.545 
.596 


.453 
.546 


.398 
.439 
.479 


.356 
.392 
•429 


•314 
.346 
.378 


.286 

•315 
.344 


W. G. 


9 


ID 


XX 


12 


13 


14 


15 


16 


17 


18 


19 


Inch. 


.148 

9/64/ 

lbs. 


.134 

9/64^. 


.120 

yib. 


.109 

7/64 


.095 

3/3a/ 


.083 

s/64/ 


.072 
5/64 b. 


.065 


.058 
x/x6 b. 


.049 
3/64/ 


.042 

3/64*. 


DiAM. 
inches. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 




1.90 

2.II 

2.33 

2.54 


1.74 

1.93 

2.13 
2.32 


'•5f 
1.76 

1.94 
2.12 


1.45 
1.60 

1.76 

1.92 


1.28 
1. 41 

1.55 
1.69 


113 
1.25 

1-37 
1.49 


.986 
.991 
1.20 

1.30 


.896 
.991 
1.09 
1. 18 


.804 
.888 

•972 
1.06 


.684 

•755 
.827 

.898 


.590 
.651 
.712 

.773 


2 

2>^ 


a. 76 
2.97 

319 
3-40 


2.52 
2.71 
2.91 

3.10 


2.30 

2.47 
2.65 

2.83 


2.08 
2.24 

2.39 
2.55 


1.83 
1.97 
2.10 
2.24 


1. 61 

1.85 
1.97 


1.40 

1. 61 
1.72 


1.28 

^•37 

1.56 


1.14 
1.23 

1-31 
139 


.969 
1.04 
i.ii 
1.18 


.834 

•895 
.956 

1.02 


2X 
2^ 
2^ 


3.62 

3.83 
4.04 


3.30 
3.49 
3.69 


3.01 

319 

3.37 


2.71 
2.86 
3-02 


2.38 
2.52 

2.66 


2.09 
2.21 

2.33 


1.82 

1.93 
2.03 


1.66 

1.85 


1.48 

1.56 
1.65 


1.25 

133 
1.40 


1.08 

1. 14 
1.20 



WEIGHT OF BRASS TUBES. 



267 



Table No. 95 (continued). 

Length, i Foot. Thickness by Holtzapffel's Wire-Gauge (Table No. 13). 

Specific Weight = I. H (Specific Weight of Wrought Iron=i). 



Thick- 
























NBSS. 

W. G. 


3 


4 


5 


6 


7 


8 


9 


ID 


IX 


Z2 


13 




.259 


.238 


.220 


.203 


.180 


.165 


.148 


.134 


.120 


.109 


•095 


Inch. 


x/ 


'5/64/ 


7/3>/ 


'3/64 


3/16 ^. 


"/64 *• 


9/64/ 


9/64^. 


>^^. 


7/64 


3/32 


DiAM. 

inches. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


2 


5.24 


4.87 


^55 


4.24 


3.80 


3.52 


3.19 


2.91 


2.65 


2.39 


2.10 


2yi 


5.62 


5-22 


4.87 


4-54 


4.07 


3-76 


3.40 


3-«o 


2.83 


2.55 


2.24 


2H 


5-99 


5-57 


519 


4.83 


4-33 


4.00 


3.62 


3-30 


3.01 


2.71 


2.38 


2H 


6.37 


591 


551 


5.13 


459 


4.24 


3.83 


3-49 


3- 1-9 


2.86 


2.52 


^H 


6.75 


6.26 


5-83 


5.42 


4.85 


4.48 


4.04 


369 


3-37 


3- 02 


2.66- 


^^ 


7.12 


6.60 


6.14 


5.72 


5-"> 


4.72 


4.26 


3.88 


3-55 


3.18 


2.79 


2^ 


7.50 


6.95 


6.47 


6.01 


5.38 


4.96 


4.47 


4.07 


3-73 


3-34 


2.93 


^H 


7.88 


730 


6.79 


6.31 


5.64 


5.20 


4.69 


4.27 


391 


3.50 


307 


3 


8.25 


7.64 


7.II 


6.60 


590 


S-44 


4.90 


4.46 


4.09 


3.66 


3.21 


3V 


9.01 


8.33 


7.75 


7.19 


6.43 


5.92 


5.49 


4.85 


4.43 


3.98 


3.48 


3>^ 


9.76 


9.02 


8.39 


7.78 


6.95 


6.40 


6.07 


5-24 


4.78 


4-30 


3.76 


3H 


10.5 


9.72 


9.03 


8.37 


7-47 


6.88 


6.65 


'5-63 


5.12 


4.61 


4.04 


4 


"•3 


10.4 


9.67 


8.96 


8.00 


7.36 


7.24 


6.02 


5.46 


4.93 


4.31 


4^ 


12.0 


ii.i 


10.3 


955 


8.52 


7.83 


7.82 


6.41 


5.80 


5.25 


4.59 


^H 


12.8 


11.8 


10.9 


10. 1 


9.04 


8.31 


8.41 


6.80 


6.15 


5§S 


4.87 


aH 


13.5 


12.5 


II.6 


10.7 


9.56 


8.79 


8.99 


7.19 


H^ 


5.88 


514 


5 


14.3 


13-2 


12.2 


"3 


10. 1 


9.27 


9-57 


7.58 


6.83 


6.20 


5.42 


5^ 


^^•?. 


X3-9 


12.9 


11.9 


10.6 


9.75 


10.2 


7.97 


7.17 


6.51 


5.69 


^'4 


15.8 


14.6 


13.5 


12.5 


II.I 


10.2 


10.7 


8.36 


7.52 


6.83 


5-97 


sH 


16.5 


15-3 


14. 1 


131 


II. 7 


10.7 


"3 


8.75 


7.86 


7.1S 


6.25 


6 


17.3 


159 


14.8 


13.7 


12.2 


II. 2 


11.9 


9.14 


8.20 


7.46 


6.52 



268 



WEIGHT OF METALS. 



Table No. 96. — Weight of One Square Foot of Sheet Brass. 

Thickness by Holteapffeirs Wire-Gauge (Table No. 13). 





Weight of 






Weight of 






Weight of 


Thickness. 


I Square 


Thickness. 


X Square 


Thickness. 


T Square 




Foot. 






Foot. 






Foot. 


No.W.G. 


inch. 


pounds. 


N0.W.G. 


inch. 


pounds. 


N0.W.G. 


■inch. 


pounds. 


3 


•259 


10.9 


II 


.120 


5-05 


19 


.042 


1.77 


4 


.238 


lO.O 


12 


.109 


4.59 


20 


•035 


1.47 


5 


.220 


9.26 


13 


•095 


4.00 


21 


.032 


1-35 


6 


.203 


8.55 


14 


.083 


3-49 


22 


.028 


i.i8 


7 


.180 


7.58 


15 


.072 


3.03 


23 


.025 


1-05 


8 


.165 


6.95 


16 


.065 


2.74 


24 


.022 


.926 


9 


.148 


6.23 


17 


.058 


2.44 


25 


.020 


.842 


10 


.134 


SM 


18 


.049 


2.06 









Table 97. — Size and Weight of Tin Plates. 



Mark. 


Size of Sheets. 


Number 
of Sheets 
in a Box. 


Weight 
per Sox. 


IC 

IX 

IXX 

IXXX 

IXXXX 


inches, inches. 
14 X 10 

» >l 
if >» 
it If 


sheets. 
225 

>» 
99 


pounds. 
112 
140 
161 
182 
203 


SDC 
SDX 
SDXX 
SDXXX 
S D XXXX 


15 X II 

>f 9f 
» if 
» » 


200 


168 
189 
210 
231 
252 


DC 

DX 

DXX 

DXXX 

DXXXX 


17 X 12}^ 


100 


98 
126 

147 
168 

169 



WEIGHT OF TIN AND LEAD PIPES. 



269 



Table No. 98. — ^Weight of Tin Pipes, 

As manufactured. 



I FOOT IN LENGTH. 



Diameter 
Externally. 



inches. 
1/ 



n 

H 



1 1 



■i 

2 



Thickness. 



3/32" inch. 



lbs. 
.148 

.384 
.620 

.856 

1.095 

1.328 

1.564 

1.802 



^ inch. 



lbs. 

.472 

.787 
T.IO3 
I.417 

1-732 
2.047 

2.362 



Diameter 
Externally. 



inches. 
2l' 



2^ 

3 



Thicknbss. 



}i inch. 



lbs. 

5-04 

5-67 
6.30 

6.93 
7.56 
8.19 



Table No. 99.— Weight of Lead Pipes. 

As manufactured. 



! 
Bore. 


Length. 


Weight and Thickness of Pipe. 




Calcu- 




Calcu- 




Calcu- 




Calcu- 






Weight. 


lated 
Thick- 
ness. 


1 Weight 


lated 
Thick- 
ness. 


Weight. 


lated 
Thick- 
ness. 


Weight 


lated 

Thick- 

ness. 


inches. 


feet 


lbs. 


inch. 


lbs. 


inch. 


lbs. 


bch. 


lbs. 


inch. 


>^ 


IS 


14 


.097 


16 


.112 


18 


.124 


22 


.146 


^ 


n 






17 


.101 


21 


.121 


30 


.140 


Va 


w 


24 


.112 


28 


.147 


32 


.181 


36 


•215 


I 


» 


36 


.136 


1 42 


.156. 


56 


.200 


64 


.225 


^Va 


12 


36 


.139 


42 


.160 


48 


.180 


52 


•193 


^% 


» 


48 


.156 


56 


.179 


72 


.224 


84 


.257 


^H 


w 






72 


.199 


84 


.228 


96 


.256 


2 


» 


72 


.178 


84 

1 


.204 


96 


.231 


112 


.266 


2^ 


10 






i 84 


.200 


96 


.227 


112 


.261 


3 


» 










112 


.218 


140 


•275 


3J^ 


» 










130 


.225 


160 


•273 


4 


» 










170 


.257 


220 


•327 


4j^ 


19 










170 


.232 


220 


•295 


4H, 
4'A 


V.6ii 


ich thick. W< 


sight pel 

9J 


r lineal f 


bot 




.... 22.04. lbs. 


99 








.... 23.25 ff 


4H 
5 


















99 


99 








.... 2*1. ( 


56 „ 


99 










-TV jy 



270 



WEIGHT OF METALS. 



Table No. loo. — Dimensions and Weight of Sheet Zinc. 

( Vielie-Montagne.) 













SiZBS OP 


Shebts. 






Weight 














No. 


Thickn«»« 


9.0 X. 50 


metres; 


3.0 X. 65 


metres; 


9. ox. 80 metres; 


per 








area, x square metre. 


area, 1.3 sq., metres. 
6. 56 X 2. X3 feet; area. 


area, 1.6 sq. metres. 
6 '56X9. 62 feet; area. 


square 
toot. 


6. 56 X X. 64 feet ; area, 








X0.76 square feet. 


13.99 square feet. 


X7. 29 square feet. 




No. 


inxn. 


inch. 


kUs. 


lbs. 


kUs. 


lbs. 


kils. 


lbs. 


lbs. 


9 


.41 


.0161 


2.90 


6.39 


3.70 


8.16 


4.6 


10.14 


.589 


lO 


•51 


.0201 


3-45 


7.61 


4.45 


9.81 


5.5 


12.12 


.704 


II 


.60 


.0236 


4.05 


8.93 


5.30 


11.68 


6.5 


14.33 


.832 


12 


.69 


.0272 


4.65 


10.25 


6.10 


13.45 


7.5 


16.53 


.960 


13 


.78 


.0307 


5-30 


11.68 


6.90 


15.21 


8-5 


18.74 


1.088 


14 


.87 


•0343 


5.95 


13-12 


7.70 


^5*§^ 


9-5 


20.94 


I.216 


^5 


.96 


.0378 


6.55 


14.44 


8.55 


18.85 


10.5 


23.15 


1.344 


i6 


1. 10 


.0433 


7.50 


16.53 


9-75 


21.50 


12.0 


26.46 


1.536 


17 


1.23 


.0485 


8.45 


18.63 


10.95 


24.14 


13.5 


29.97 


1.740 


i8 


1.36 


.0536 


9-35 


20.61 


12.20 


26.90 


15.0 


33.07 


1.920 


19 


1.48 


.0583 


10.30 


22.71 


13.40 


29.54 


16.5 


36.38 


2.II2 


20 


1.66 


.0654 


11.25 


24.80 


14.60 


32.19 


18.0 


39.68 


2.304 


21 


1.85 


.0729 


12.50 


27.56 


16.25 


35-82 


20.0 


44.09 


2.560 


22 


2.02 


.0795 


13.75 


30.31 


17.90 


39.46 


22.0 


48.50 


2.816 


23 


2.19 


.0862 


15.00 


33-07 


19.50 


42.99 


24.0 


52.91 


3.073 


24 


2.37 


•0933 


16.25 


35.82 


21.10 


46.52 


26.0 


57.32 


3.329 


^5 


2.52 


.0992 


17.50 


38.58 


22.75 


50.15 


28.0 


61.73 


3.585 


26 


2-66 


.1047 


18.80 


41.44 


24.40 


53.79 


31.0 


68.34 


3.969 



Table No. loo {continued). 
Special Sizes for Sheathing Ships. 









Sizes op 


Sheets. 


Weight 








No. 


Thickness. 


X.1SX.35 metres; 


1.30 X. 40 metres; 


per 






area, .402 sq. metre. 

] 3.77 X X. 15 feet; area, 
4-33 sq- feet. 


area, .520 sq. metre. 

4. 26X1. 31 feet; area, 
5.60 sq. feet. 


square 
foot. 


No. 


mils. 


inch. 


i kUs. 


lbs. 


kils. 


lbs. 


lbs. 


15 


.96 


.0378 


2.65 


5.84 


3.40 


W 


1.344 


16 


1. 10 


.0433 


3.00 


6.61 


3.90 


8.60 


1.536 


17 


1.23 


.0485 


3.40 


7.50 


4.40 


9.70 


1.740 


18 


1.36 


■0536 


3.75 


8.27 


4.90 


10.80 


1.920 


19 


1.48 


.0583 


, 4.15 


9.15 


5.35 


11.79 


2.II2 


20 


1.66 


.0654 


; ^55 


10.03 


5.85 


12.90 


i 2.304 

1 



Note. — A deviation of 25 dekagrammes, or about half-a-pound, more or less, from the 
proper weight of each number of sheet, is allowed. 

Nos. I to 9 are employed for perforated articles, as sieves, and for articles de Paris, 
Nos. 10 to 12 are used in the manufacture of lamps, lanterns, and tin-ware generally, and 
for stamped ornaments. The last numbers are used for lining reservoirs, and for baths 
and pumps. 



FUNDAMENTAL MECHANICAL PRINCIPLES. 



FORCES IN EQUILIBRIUM. 
Solid Bodies. 




Paralldogram of Forces. — ^When a body remains at rest whilst being acted 
on by two or more forces, it is said to be in a state of equilibrium, and so also 
are the forces. Thus, if the forces p/, q ^, r r, Fig. 86, acting on the body 
pqr^ keep it at rest, they are in 
equilibrium, and any two of them 
balance the third. The lines of 
force, if produced, meet at one point 

o within the body, and if a parallel- r 

(^ram be constructed having two 
adjacent sides proportional to and 

parallel to two of the forces respec- Fig. 86.— EquiUbrium of Forces. 

tively, to represent them in magni- 
tude and direction, the diagonal of the parallelogram will represent the third 
force in magnitude and direction. Let the lines o p, OQ, Fig. 87, represent 
the forces p/, q^ in magni- 
tude and direction, and com- :p 
plete the parallelogram • by 

drawing the parallels p r, q r, Bf_„ 

and draw o r. Then o r re- 
presents in magnitude and 
direction the resultant of the ^ 

two forces; and RO taken in Fig. 87.— Parallelogram of Forces. 

the opposite direction repre- 
sents the third force Rr, Fig. 86. If it be applied in this direction to 
the point o, as indicated by a dot line o r', it would balance the other two. 
This construction is called the paralldogram of forces y and is applicable to 
any three forces in equilibrium. 

Three forces in equilibrium may also be represented by a triangle, or 
half a parallelogram. For example, the triangle o p r represents by its 
three sides the forces o P, o q, o r, the side p r being substituted for o q. 

Three forces in equilibrium must be in the same plane. 

When the directions of three forces holding a body at rest, and also the 
magnitude of one of them, are known, the magnitudes of the other two can 
be determined by constructing a parallelogram in the manner above exem- 
plified, and measuring the lengths of the 5ide^ apd the diagonal. 




2/2 



FUNDAMENTAL MECHANICAL PRINCIPLES. 



Polygon of Forces, — Equilibrium may subsist between more than three 
forces, which need not necessarily be in the same plane; and they can, like 
those already illustrated, be developed in direction and magnitude by diagram. 
Thus, let the point o, Fig. 88, representing a solid body, be kept at rest by a 
number of forces, op, o q, o r, o s, o t. Find the equivalent diagonal o/ for 
the first two forces ; then construct the parallelogram and diagonal o r for the 
resultant of op and the third force or; and again the parallelogram and 
diagonal o s for the resultant of o r and the fourth force o s. The last 
resultant o^ represents in one the four distributed forces op, oq, or, os, and 
it balances the fifth force o t equal and opposite to it. A5 o j and o t are 
in the same straight line, their resultant is of course nothing. 

The several forces thus dealt with may be combined into a polygon of 
forces. Draw o p. Fig. 89, parallel and equal to o p, Fig. 88, p q parallel and 

equal to o Q, Q r parallel to o r, r s 
parallel to os; then, finally, so, 
completing the polygon, will be 
parallel and equal to ot. Fig. 88, 
the last of the series. Professor 
Mosely illustrates the polygon of 
forces by the united action of a 
number of bell-ringers, pulling by a 
number of ropes joined to a single 
rope. The polygon constructed as 
in Fig. 90, shows successively by 
corresponding letters, the individual 
contributions of the bell-ringers, 
combined into one vertical force. 

Again, equilibrium may be estab- 
lished between a number of forces 




Fig. 88.— Equilibrium of more than Three Forces. 



acting in the same plane, applied to different points in a body, or system of 
bodies. For example, let the forces p o, q o, r o, s o, and t o, be applied 

to several points, o, o, o, o, o, on a flat board 
ABC, Fig. 91, by means of cords and weights; 
it will settle into a position of equilibrium, 
when the opposing forces arrive at a balance 
between themselves. An axis or pivot may 
be established at any point, m, on the surface 
of the board, without disturbing the equilib- 
rium, and it may be viewed as a centre of 
motion round which the forces tend to turn 
the board, some in one direction, the others 
the opposite way, balancing each other. The 
effect of each force to turn the body about 
the centre is represented by the product of its magnitude by the leverage^ 
or perpendicular distance of its direction from the centre; draw these 
perpendiculars, and multiply each force by its perpendicular or leverage, 
then the resulting products will be divisible into two sets, tending to turn 
the board in opposite directions. The sum of the first set of products is 
equal to the sum of the second set, as is proved by the fact of equilibrium. 
Moments of Forces, — ^The product of a force by the perpendicular dis- 
tance of its direction from any given point, is called the momait of the 




Fig. 89. — Polygon of Forces. 



FORCES IN EQUILIBRIUM. 



273 




force about that point; and the equilibrium above discussed, in connection 

with Fig, 91, is the result of the equality of moments. 

The law of the equality of moments may be thus set forth; — If any 

number of forces acting anywhere in the 

same plane, on the same body or connected 

system of bodies, be in equilibrium, then 

the sum of the forces tending' to turn the 

system in one direction about any point in 

that plane, is equal to the sum of the mo- 
ments of those forces tending to turn the 

system in the other direction. 

Such balanced forces may be transferred 

to a single point, and placed about it, as in 

l~ig. 88, parallel to their directions as they 

stand; and they will continue in equilibrium, 

holding the point at rest. A polygon of 

the iotctspgrsl within Fig. 90, may be 

constructed similarly to Fig. 89. 

Though the principle of the polygon of 

forces be sufficient to test the equilibrium 

of a system of forces acting at one point, 

yet the principle of the equality of moments, Fig. 90— Beu-nngen, Polygon of Forcei. 

in addition, is necessary to establish the 

equilibrium of a system applied to different points. The two principles 

conjointly are necessary, and they are sufficient, as conditions of equilibrium. 
The Catenary. — When 

a chain, or a rope, or a 

flexible series of rods, is 
suspended by its extremi- 
ties, supporting weights 
distributed along its 
length, in a state of rest, 
ii forms a polygon of 
forces in equilibrium, as 
in Fig. 92. If all the 

forces except those which o 

act on the extremities of 
the chain, be combined 
into a resultant, then the 
two extreme sides being 
produced, will meet the 
direction of the resultant 
at one point Thus, in 
the polygon, Fig. 92, 
loaded with weights, w,w, 

&C., the verrical resultant Fig. 91.— Equjiiry of Moments. 

of these weights w' W, 

passing through their common centre of gravity, intersects at w' the two 
extreme sections p a, p' b, when produced downwards. 

Similarly, in the catenary, Fig. 93, which is the curve assumed by a rope 
or other flexible medium uniformly loaded and suspended by the two 



274 



FUNDAMENTAL MECHANICAL PRINCIPLES. 



extremities, if tangents be drawn to the extremities a, b, of the curve, meeting 
at w*, they represent the directions of the forces sustaining the curve at 





Fig. 9a.— The Catenary. 



Fig. 93. — ^The Catenary. 



those points, and they intersect at the same point w*, the vertical line G w^ 
passing through tlie centre of gravity of the curve. Let the weight of the 




•^ 



flf 



T 



Fig. 94.— Centrifugal Forces in Equilibrium. 



Fig. 95.—- Parallel Forces in Equilibrium. 



curve be represented by g w^, and complete the parallelogram m n, then 
w* M and w*' n represent in force and direction the tension at the points 
B and A. 

Centrifugal Forces in Equilibrium. — ^\Vhen a cylindrical vessel is exposed 
to a uniform internal pressure, as the pressure of steam within a boiler, for 
example, the pressure is balanced by the resistance of cohesion of the 
material of the boiler. Let a b c D, Fig. 94, be the section of a cylindrical 
boiler, the radial pressure of the steam may be represented by the arrows, 
which are equal and opposite in direction. The tension on the metal in 
resisting the internal pressure at any particular section b i>, is equal to the 
sum of the pressures resolved into the direction at right angles to b d, or 
parallel to ac, according to the triangles, or half-parallelograms of force 
attached to each oblique arrow. The total vertical pressure thus obtained 
by the resolution of forces is equal to the total vertical pressure which 



FORCES IN e;quilibrium. 



27s 



would be exerted on the sectional line B d if it be supposed to be a rigid 
diaphragm across the boiler, which is easily calculated. If the radial 
pressure be, for example, 100 lbs. on each square inch of surface, then the 
total pressure, or the tension on the two sides at b and d, would be 
100 X BD on each inch of length of the two sides; that is to say, if the 
diameter b d be equal to 60 inches, the tension on the two sides would be 
60 X 100=6000 pounds for each inch of length. 

A similar argument applies to the tension on the rim of a revolving fly- 
wheel. 

Parallel Farces. — Systems of parallel forces are particular cases of the 
forgoing. — Let a, b, c, d, e, f, Fig. 95, be a system of parallel forces in equili- 
brium; and MN a line perpendicular to them in the same plane, and cut 
by them at the points a^byCyd^e,/, They may act at any points in their 
lines of direction without disturbing the equilibrium, and they may be sup- 
posed to be applied at those points in the line m n. Then, the sum of the 
moments of the three forces a, b, c, acting in one direction, with respect to 
any point m as a centre, is equal to the sum of 
the moments of the forces d, e, f, opposed to 
them. Further, the sum of the simple forces 
A, B, c, irrespective of their moments, is equal to 
that of the forces d, e, f, so that the fact of their 
being in equilibrium resolves itself into a case of 
action and reaction, for the two equivalent forces 
representing the two opposing sums, act in the 
same straight line in opposite directions. 

When three parallel forces balance each other, 
acting on a straight line, two of them must be 
opposed to the third; and the third must act 
between the other two, being equal and opposite 
to their resultant. Let a, b, c, Fig. 96, be three 
such forces applied to the line e g f, at the points 
E, G, f respectively; then, with respect to the 
point G, the moment of the force b is nothing, because it passes through 
that point and has no leverage on it There remain the moments of the 
extreme forces, a and c, which are equal to each other, that Is to say 

axeg = cxfg. 




Fig. 96.— Three Parallel Forces in 
Equilibriuia. 



From this it follows, by proportion, that 

A : c : : FG : eg, 

and that the extreme forces are to each other inversely as their distances 
from the middle force. 

In general, of three parallel forces acting in equilibrium on an inflexible 
line, the first in order is to the third as the distance of the third from the 
second, is to that of the first from the second. 

The sum of the first and third is equal to the second; and when the 
distances or leverages are equal, the first and third forces are equal to each 
other. 

If the position of the line e f be inclined to the direction of the three 
forces, and changed to e' f'. Fig. 96, the forces A, b, c, continue in equilibrium; 



2'j(> 



FUNDAMENTAL MECHANICAL PRINCIPLES. 




Fig. 97. — Parallelopiped of Forces. 



for the perpendicular lines g e and g f continue, as before, to be the lever- 
ages of the extreme forces a and c, on the central point g. 

When three forces not in the same plane act on one point, there cannot 
be equilibrium between thenL Two of these may be reduced to their 

resultant by parallelogram, and 
this resultant reduced with the 
third force to a final resultant 
For example, let the lines op, 
OQ, OR, Fig. 97, represent in 
magnitude and direction three 
forces not in one plane acting 
on the point o. By parallel- 
ogram, o s is the resultant of the 
two forces o p, o Q, and o t is the 
final resultant of o s and the third 
force o R. That is to say, o t is the resultant of the three given forces. 

If parallelograms be formed from each two of the three forces, they form, 
when duplicated, a parallelopiped of forces, of which the diagonal is found 
by the final resultant o t, and the principle of the parallelopiped of forces 
may be thus defined: — If three forces be represented in magnitude and 
direction by three adjacent edges of a parallelopiped, their resultant is 
represented in magnitude and direction by the adjacent diagonal of the 
solid. 

There must be at least four forces to produce equilibrium about a point, 
when the forces are not in the same plane. 

The triangle ost. Fig. 97, comprises in its three sides the resultant of 
the first and second forces, the third force, and the resultant of the three. 
If the first resultant o s be replaced by the two lines o Q and Q s, which 
represent the first and second forces, they form the four-sided figure o Q s t, 
the polygon of the four equilibrating forces. 

A greater number of forces than four acting on a point may be reduced 
in like manner. 

Fluid Bodies. 

The characteristic property of fluids is the capability of transmitting the 
pressure which is exerted upon a part of the surface of the fluid, in all 
directions, and of the same intensity: — the same pressure per square inch or 
per square foot. 

The pressure of water in a vessel, caused by its own gravity, increases in 
proportion to the depth below the surface; and the pressure on a horizontal 
surface, say, the bottom, is equivalent to the weight of the superincumbent 
column of water, and the intensity of the pressure is independent of the 
form of the vessel. The same rule applies when the pressure is from below 
upwards. 

The same rule also applies when the surface is either vertical or inclined, 
and the mean height of the superincumbent column of water is measured 
by the depth of the centre of gravity of the given surface below the siurface 
of the water. 

The water in open tubes communicating with each other, when in a state 
of equilibrium, stands at the same level in the tubes, whatever may be the 
relative diameters of the tubes. 



MOTION. — GRAVITY. 277 

The height of the superincumbent column of water is called the head of 
water. 

The buoyancy^ or the upward force with which water presses a body- 
immersed in it, from below upwards, is equal to the weight of water dis- 
placed by the body, or of a quantity of water equal in volume to the sub- 
merged body, or submerged portion of a body. The resultant of the 
upward pressure passes through the centre of gravity of the water displaced; 
and also, when the floating body is at rest, through the centre of gravity of 
th« body. 

This resultant line is called the axis of floatation, and the horizontal 
section of the body at the surface of the water is \ht plane of floatation. 

Bodies float either in an upright position or in an oblique position. A 
body floats with stability , when it strives to maintain the position of equili- 
brium, and when it can only be moved out of this position by force, and 
will return to it when the force is withdrawn. The metacentre is the point 
at which the axis of floatation intersects the axis of a symmetrical body, as a 
ship, when inclined. If the metacentre lies above the centre of gravity of 
the ship, the ship floats with stability; if below, the ship is unstable; if the 
centres coincide, which they must do in a cylinder or a sphere, for example, 
the body floats indifferently in any position. 

For the weight, volume, and pressure of water and air, see Water and 
Air as standards of measure^ page 124. 

MOTION. 

The motion of a body is uniform, when the body passes through equal 
spaces in equal intervals of time. 

Velocity is the measure of motion, and is expressed by the number of 
feet or oflier unit of length moved through per second or other unit of time. 

Motion is accelerated, when the body moves through continually increased 
spaces in equal intervals of time, like a railway train starting from a station. 
Motion is retarded, when the body moves through continually decreased 
spaces in equal intervals of time, like a railway train arriving at a station. 
The acceleration and retardation are uniform, when the spaces moved through 
increase or decrease by equal successive amounts, like a body falling by 
the action of gravity, or, on the contrary, projected upwards in opposition 
to gravity. 

GRAVITY. 

When bodies fall freely near the surface of the earth, the motion, as 
already said, is uniformly accelerated; equal additions of velocity being made 
to the motion of the body in equal intervals of time. 

During the ist second of time, the body, starting from a state of rest, 
falls through 16.095 ^^^t, or, say 16.1 feet; during the 2d second, it 
falls through three times 16.1 feet; during the 3d second, it falls through 
five times 16.1 feet, and so oa The body having, in the ist second, fallen 
through 16. 1 feet, from a state of rest, with a motion uniformly accelerated, 
it would move through 32.2 feet in the next second, with the velocity 
thus acquired, without any additional stimulus from gravity; that is to 
say, the velocity acquired at the end of the ist second is 32.2 feet per second. 
During the 2d second, it in fact acquires an additional velocity of 32.2 feet 
per second, making up, at the end of this second, a final velocity of 64.4 



2/8 FUNDAMENTAL MECHANICAL PRINCIPLES. 

feet per second. In like manner the body acquires an additional velocity 
of 32.2 feet per second during the 3d second, making a final velocity of 
three times 32.2 feet, or 96.6 feet per second. And so on. 

Each of these additional velocities is acquired in falling through 16.1 feet 
additional to the space fallen through in virtue of the velocity acquired at 
the beginning of each second. 

The relations of height fallen^ velocity acquired^ and time of fally are 
simply exhibited in the following manner : — 

During the successive seconds the heights fallen through are consecutiv|?ly 
as follow: — 

time, I, I, I, I second, 

height of fall, 16. i, 16. i x 3, 16. i x 5, 16.1 x 7 feet 

And reckoning the totals from the commencement of the fali, 

total times, i, 2, 3, 4 seconds, 

total height of fall^ 16.1, 16.1 x 4, 16. i x 9, 16. i x 16 feet. 

or t6.i, 16. 1 X 2^, 16. 1 X 3^, 16. i x 4^ feet. 

or 16.1, 64.4, i44'9> 257.6 feet 

Showing that the total height fallen is as the square of the total time. 

Again, during the successive seconds, the successive additional velocities 
acquired are : — 

time, I, I, I, I second, 

velocities acquired, 32.2, 32.2, 32.2, 32.2 feet per second 

And the total or final velocities acquired, reckoning from the commence- 
ment of the fall, are : — 

total times, i, 2, 3, 4 seconds, 

final velocities, 32.2, 32.2 x 2, 32.2 x 3, 32.2 x 4 feet per second. 

or 32.2, 64.4, 96.6, 128.8 feet per second. 

m 

Showing that the velocity acquired is in direct proportion to the time of the 
fall. 

The above relations of time, height, and velocity are brought together 
for comparison, thus : — 

time as, i, 

velocity acquired as, i, 
height of fall as, i, 

or as I, 

Showing that, whilst the velocity increases simply with the time, the lieight 
of fall increases as the square of the time, and as the square of the velocity. 

The force of gravity is expressed by the velocity communicated by gravity 
to a body falling freely in a second, namely, 32.2 feet per second, and is 
symbolized by g. 

The foregoing relations of time^ velocity^ and height of fall, are comprised 
in the six following propositions with their answers — applicable to the 
condition of a body falling freely. They are much used in mechanical 
calculations. 

I and 2. Given the time^ to find the velocity and the height, 
3 and 4. Given the velocity ^ to find the time and the height, 
5 and 6. Given the height, to find the time and the velocity. 



2, 


Zy 


4, &C. 


2| 


3» 


4, &C 


4, 


9» 


16, ^C. 


2^ 


6 y 


42, &c. 



GRAVITY. 279 

Rules for the Action of Gravity. 

Putting /=the time of falling in seconds, v = the velocity in feet per 
second, ^=the height of fall in feet, and ^= gravity or 32.2; then 

Rule i. Given the time of fall, to find the velocity acquired by a falling 
body. Multiply the time in seconds by 32.2, the product is the final 
velocity in feet per second. Or 

Z/ = 32.2 / ( I ) 

Rule 2. Given the time of fall, to find the height of the fall. Multiply 
the square of the time in seconds by 16.1. The product is the height of 
fall in feet. Or 

^=16.1 /2 (2) 

Rule 3. Given the velocity^ to find the time of falling. Divide the 
velocity in feet per second by 32.2. The quotient is the time in seconds. 
Or 

/=-^ (3) 

32.2 

Rule 4. Given the velocity^ to find the height of fall "due" to the 
velocity. Square the velocity in feet per second, and divide by 64.4. The 
quotient is the height of fall in feet Or 

/i=g^ (4) 

64.4 

Rule 5. Given the height of fall, to find the time 6f falling. Divide the 
height in feet by 16.1, and find the square root of the quotient The root 
is the time in seconds. Or 



/ 



or / 



= K\/ h (s) 



Rule 6. Given the height of fall, to find the velocity due to the height 
Multiply the height in feet by 64.4, and find the square root of the product 
The root is the velocity in feet per second. 

Or, multiply the square root of the height in feet by 8.025; the product 
is the velocity in feet per second. 

Note. — It is usual to take the integer 8 only for the multiplier. 

Sjonbolically, these operations are expressed as follows : — 

z/ = 32.2\/-^ — = a/64.4^ = 8.025 a/ ^ 
j^ 32.2 ^ ^ 

OT in a round number z^= 8 a/ h '. (6) 

The above rules are applicable, inversely, to the motion of bodies pro^ 
jected upwards and uniformly retarded by gravity. The height to which a 
body projected vertically upwards by an initial impulse, will ascend, is equal 
to tie height through which the body must fall in order to acquire the 
initial velocity, and the same rule (4) applies in these two cases. 



28o 



FUNDAMENTAL MECHANICAL PRINCIPLES. 



The following table, No. loi, gives the velocity acquired by a falling body 
in falling freely through a given height Table No. 102 gives, conversely, 
the height of fall due to a given velocity. Table No. 103 gives the fall and 
the final velocity due to a given time of falling freely. 



Table No. loi. — ^Velocity acquired by Falling Bodies, due to Given 

Heights of Fall. 



r^= 8.025 >y/ ^- 



Hdghtof 
Van 


Vdodty 

in Feet 

per Second. 


Height of 
FalL 


Velocity 

in Feet 

per Second. 


Height of 
FalL 


Velocity 

in Feet 

per Second. 


Hdgfatof 


Vdodty 

in Feet 

per Second. 


feet. 


ft. per sec. 


feet 


ft. per sec 


feet. 


ft. per sec 


feeL 


ft. per sec 


.01 


.803 


3-0 


13.90 


23 


38.49 


50 


56.74 


.02 


1. 14 


3-5 


15.01 


24 


39.31 


100 


80.25 


.03 


1-39 


4.0 


16.05 


25 


40.12 


150 


98.28 


.04 


1.61 


4.5 


1703 


26 


40.92 


200 


"35 


.05 


1.80 


5-0 


17.99 


27 


41.70 


300 


139.0 


.06 


1.97 


5-5 


18.82 


28 


42.47 


400 


160.5 


.07 


2.12 


6.0 


19.66 


29 


43.22 


500 


179.9 


.08 


2.27 


6.5 


20.46 


30 


43.95 


600 


196.6 


.09 


2.41 


7.0 


21.23 


31 


44.68 


700 


212.3 


.1 


2.54 


7.5 


21.97 


32 


45.39 


800 


226.9 


.2 


3.20 


8.0 


22.69 


33 


46.10 


900 


240.7 


•3 


4.40 


8.5 


23.40 


34 


46.79 


1000 


253-8 


.4 


5'07 


9.0 


24.07 


35 


47.47 


1500 


310.8 


•5 


5.68 


9-5 


2473 


36 


48.15 


2000 


3589 


.6 


6.22 


10 


25.38 


37 


48.81 


2500 


401.2 


.7 


6.71 


II 


26.62 


38 


49.47 


3000 


439-5 


.8 


7.18 


12 


27.80 


39 


50.11 


3500 


474-7 


•9 


7.61 


13 


28.93 


40 


50.75 


4000 


507.S 


I.O 


8.03 


14 


30-03 


41 


51.38 


4500 


538.3 


1.2 


8.79 


15 


31.08 


42 


52.01 


5000 


567-4 


L4 


9.50 


16 


32.10 


43 


52.62 


6000 


621.6 


1.6 


10.15 


17 


33.09 


44 


53.23 


7000 


671.4 


1.8 


10.77 


18 


34.05 


45 


53.83 


8000 


717.8 


2.0 


"35 


19 


34.98 


46 


54-43 


9000 


761.3 


2.25 


12.04 


20 


35.89 


47 


55.02 


1 0000 


802.5 


2.50 


12.69 


21 


36.77 


48 


55.60 






2.75 


13-31 


22 


37.64 


49 


56.17 







GRAVITY. 



281 



Table No. 102. — Height of Fall due to Given Velocities. 



h^ 



64.4" 



Velocity 
in Feet 


Height of 
Fall. 


Velocity 
in Feet 


Height of 

Fall 


Velocity 
in Feet 


Height of 
Fall 


Velocity 
in Feet 


Height of 
Fall. 


per Second. 


per Second. 


per Second. 


per Second. 


fc per sec 


feet. 


(t per sec. 


feet 


ft per sec 


feet 


ft. per sec 


feet 


•25 


.0010 


19 


5.61 


46 


32.9 


73 


82.7 


•50 


.0039 


20 


6.21 


47 


34.3 


74 


85.0 


■75 


.0087 


21 


6.85 


48 


35.8 


75 


87.4 


1. 00 


.016 


22 


7.52 


49 


37.3 


80 


99.4 


1-25 


.024 


23 


8.21 


50 


38.8 


85 


II2.2 


1-50 


.035 


24 


8.94 


51 


40.4 


90 


125.8 


1-75 


.048 


25 


9-71 


52 


42.0 


95 


140. 1 


2 


.062 


26 


10.5 


53 


43-6 


JOG 


155-3 


2-5 


.097 


27 


II-3 


54 


45-3 


105 


171. 2 


3 


.140 


28 


II. 2 


55 


47.0 


no 


187.9 


3-5 


.190 


29 


131 • 


56 


48.7 


115 


205.4 


4 


.248 


30 


14.0 


57 


50-4 


120 


223.6 


4-5 


.314 


31 


14.9 


58 


52.2 


130 


262.4 


5 


.388 


32 


159 


59 


54.1 


140 


304.3 


6 


.559 


33 


16.9 


60 


55-9 


150 


349-4 


7 


.761 


34 


17.9 


61 


57.8 


175 


475-5 


8 


.994 


35 


19.0 


62 


59.7 


200 


621 


9 


1.26 


36 


20.1 


63 


61.6 


300 


1397 


JO 


1.55 


37 


21.3 


64 


63.6 


400 


2484 


XI 


1.88 


38 


22.4 


65 


65.6 


500 


3882 


J2 


2.24 


39 


23.6 


66 


67.6 


600 


5590 


13 


2.62 


40 


24.9 


67 


69.7 


700 


7609 


J4 


304 


41 


26.1 


68 


71.8 


800 


9938 


15 


3-49 


42 


27.4 


69 


73-9 


900 


12578 


16 


398 


43 


28.7 


70 


76.1 


1000 


15528 


17 


4.49 


44 


30.1 


71 


78.3 






'* 1 


5'"^^ 


45 


31-4 


72 


80.5 







282 



FUNDAMENTAL MECHANICAL PRINCIPLES. 



Table No. 103. — Height of Fall and Velocity acquired, for 

Given Time of Fall. 

^ = 16.1 A' V-^2,2t 



' 




Velocity 






Velocity 






Velocity 


Time of 


Height of 


acquired in 
Feet per 


Time of 


Height of 


acquired in 
Feet per 


Time of 


Height of 


acquired in 
Feet per 


Fall 


Fall. 


Fall 


FalL 


Fall. 


Fall. 






Second. 






Second. 






Second. 


seconds. 


feet. 


ft. per sec. 


seconds. 


feet. 


ft. per sec. 


seconds. 


feet. 


ft. per sec 


I 


16. 1 


32.2 


12 


2318 


386.4 


23 


8517 


740.6 


2 


64.4 


64.4 


13 


2721 


418.6 


24 


9273 


772.8 


3 


144.9 


96.6 


14 


3156 


450.8 


25 


10062 


805.0 


4 


257.6 


128.8 


15 


3623 


483.0 


26 


10884 


837.2 


5 


402.5 


161. 


16 


4122 


515-2 


27 


II737 


869.4 


6 


579-6 


193.2 


17 


4653 


547.4 


28 


12622 


901.6 


7 


788.9 


225.4 


18 


5217 


579-6 


29 


13540 


933-8 


8 


1030 


257.6 


19 


5812 


611.8 


30 


14490 


966.0 


9 


1304 


289.8 


20 


6440 


644.0 


31 


15473 


998.2 


10 


1610 


322.0 


21 


7100 


676.2 


32 


16487 


1030 


II 


1948 


354.2 


22 


7792 


708.4 









ACCELERATED AND RETARDED MOTION IN GENERAL. 

The same rules and formulas that have been applied to the action of 
gravity are applicable to the action of any other uniformly accelerating 
force on a body, the numerical constants being adapted to the force. If an 
accelerating or retarding force be greater or less than gravity; that is to say, 
than the weight of the body, the constants 16.1, 32.2, and 64.4 are to be 
varied in the same proportion. 

To do this, multiply the constant by the accelerating force, and divide 
the product by the weight of the body. Let / be the accelerating force, 
and w the weight of the body, then the constant becomes 



16.1/ ^^ 32.2/ ^^ 64.4/. 



or 



or 



w 



w 



w 



(a) 



and substituting this in the formulas (i) to (6) for gravity, the following 
general rules and formulas are arrived at for the action of uniformly accel- 
erating or retarding forces. The rules are written for accelerating forces, 
but they apply by simple inversion to retarding forces also. 

General Rules for Accelerating Forces. 

The accelerating force and the weight of the body are expressed in the 
same unit of weight; and the space in feet, the time in seconds, and the 
velocity in feet per second. 

In the following rules the time during which a body is acted on by an 
accelerating force is called tAe titne; the velocity acquired at the end of the 



ACCELERATED AND RETARDED MOTION. 283 

time is called the final velocity; the space traversed by the body during the 
time is called the space; the accelerating force is called the force, 

t - the time. 

V = the final velocity. 

s = the space. 

/ = the force. 

w = the weight. 

Rule 7. Given the weig/it^ the force^ and the time; to find the final 
velocity. Multiply the force by the time and by 32.2, and divide by the 
weight The quotient is the final velocity. Or 

Rule 8. Given the weighty the forccy and the time; to find the space. 
Multiply the force by the square of the time and by 16. i, and divide by the 
weight Or 

. = 1^:11^ (8) 

Rule 9. Given the weight, the filial velocity, and the force; to find the 
time. Multiply the final velocity by the weight, and divide by the force, 
and by 32.2. The quotient is the time. Or 

. W V / X 

32.2/ 

Rule 10. Given the weight, \ht final velocity, and the force; to find the 
space. Miiltiply the weight by the square of the velocity, and divide by 
the force, and by 64.4. The quotient is the space. Or 

W 7/^ / V 

'=-e^f ('°> 

Rule ii. Given the weight, the force, and the space; to find the time. 
Multiply the weight by the space, and divide by the force; find the square 
root of the quotient, and divide it by 4. The last quotient is the time in 
seconds. Or 

^-Hyf^ (") 

Rule i 2. Given the weight, the force, and the space; to find the final 
velocity. Multiply the space by the force, and divide by the weight; find 
the square root of the quotient, and multiply by 8. The product is the final 
velocity. Or 

'7^ 



V=%J^- 



(12) 

w ' 



Rule 13. Given the weight, the space, and \he final velocity; to find the 
force. Multiply the weight by the square of the final velocity, and divide 
by the space, and by 64.4. The quotient is the force. Or 

^=6iT* <'3) 



284 FUNDAMENTAL MECHANICAL PRINCIPLES. 

Rule 14. Given the weigJit^ timcy and final velocity; to find the force. 
Multiply the weight by the square of the velocity, and divide by the space, 
and by 32.2. Or 

/=^, (X4) 

32.2 / 

Note I. When the accelerating or retarding force bears a simple ratio to 
the weight of the body, the ratio may, for greater readiness in calculation, 
be substituted in the quantities (a) representing the modified constants, for 
the force and the weight. Suppose the accelerating force is a tenth part of 
the weight, then the ratio is i to 10, and 

16. 1 , 
__ = 1. 61, 

10 

^2.2 

^ =3.22, 



10 

644 _ 
10 



6.44; 



and these quotients may be substituted for 16. i, 32.2, and 64.4 respectively 
in the formulas for the action of gravity (i) to (6), to fit them for direct 
use in dealing with an accelerating force one-tenth of gravity, the height h 
in those formulas, of course, being taken to mean space traversed. 

Note 2. The tables, Nos. 101-103, pages 280-282, for the relations of 
the velocity and height of falling bodies, may be employed in solving 
questions of accelerating force generally. 

Example, A ball weighing 10 lbs. is projected with an initial velocity of 
60 feet per second on a level bowling-green, and the frictional resistance 
to its motion over the green is i lb. . What distance will it traverse before 
it comes to a state of rest? By rule 10, 

10 lbs. X 60^ ^^^ .. 

— Tu z — = 559 leet, 

I lb. X 64.4 

the distance traversed. 

Again, th^ same result may be arrived at, according to Note i, by- 
multiplying the constant 64.4, in rule 4, for gravity, by the ratio of the 
force and the weight, which in this case is -j^, and 64.4 x -j^ = 6.44. 
Substituting 6.44 for 64.4 in that rule and formula, the formula becomes 

Tj 60 
^ = T — ^-f- — = 559 feet, 
6.44 6.44 

the distance traversed, as already found. 

But the question may be answered more directly by the aid of the table 
for falling bodies (No. 102, page 281). The height due to a velocity of 
60 feet per second, is 55.9 feet; and it is to be multiplied by the inverse 
ratio of the accelerating force and the weight of the body, or ^, or 10; 

that is, 

55.9 X 10 = 559 feet, 

the distance traversed. 

If the question be put otherwise — What space will a ball move over 
before it comes to a state of rest, with an initial velocity of 60 feet per 



GRAVITATION ON INCLINED PLANES. 285 

second, allowing the friction to be i-ioth the weight of the ball? The 
answer may be given, that the friction, which is the retarding force, being 
i-ioth of the weight, that is of gravity, the space described will be 10 times 
as much as is necessary for gravity, supposing the ball to be projected 
vertically upwards to bring the ball to a state of rest. The height due to 
the velocity is 55.9 feet; then 

55.9 X 10 = 559 feet, 

the space described by the ball. 

Action of Gravity on Inclined Planes. 

If a body freely descend an inclined plane by the force of gravity alone, 
the velocity acquired by the body when it arrives at the foot of the plane, 
is that which it would acquire by falling freely through the vertical height. 
Or, the velocity is that " due " to the height of the plane. 

The time occupied in making the descent is greater than that due to the 
height, in the ratio of the length of the plane, or distance travelled, to 
the height. The time is therefore directly in proportion to the length of 
rhe plajie, when the height is the same. 

The impelling or accelerating force by gravitation acting in a direction 
parallel to the plane, is less than the weight of the body, in the ratio of the 
height of the plane to its length. It is, therefore, inversely in proportion 
to the length of the plane, when the height is the same. 

The time of descent, under these conditions, is inversely in proportion 
to the accelerating force. 

If, for instance, the length of the plane be five times the height, the time 
of making freely the descent on the plane by gravitation is five times 
that in which a body would freely fall vertically through the height; and 
the impelling force down the plane is '/j th of the weight of the body. 

Problems on the descent of bodies on inclined planes are soluble by the 
aid of the rules 7 to 14, for the relations of accelerating forces. But, 
as a preliminary step, the accelerating force is to be determined, by 
multiplying the weight of the descending body by the height of the plane, 
and dividing the product by the length of the plane. For example, let a 
body of 15 pounds weight gravitate freely down an inclined plane, the 
length of which is fiver times the height, the accelerating force is 1 5 -r 5 = 3 
pounds. If the length of the plane be 100 feet, the height is 20 feet, and 
the velocity acquired in falling freely firom the top to the bottom of the 
plane would be, by rule 12, 

z/ = 8/y/5-^^ — 2z=s^ 20 =35.776 feet per second. 
The time occupied in making the descent is, by rule 11, 

^=}i V i^-^^^^ = H\/ 500 = 5-59 seconds. 
^ ' 3 

Whereas, for a free vertical fall through the height, 20 feet, the time 
would be, by rule 5, 

t=%^ 20 = I.I 18 seconds, 
which is '/s ^ ^^ ^^ ^^^^ ^^ making the descent on the inclined plane. 



286 FUNDAMENTAL MECHANICAL PRINCIPLES. 

Special Rules for the Descent on Inclined Planes. 

The height and the length of an inclined plane may be substituted 
for the accelerating force and the weight respectively in the rule (ii), 
to find the time. Putting ^ = the height bf the plane, and /= the length 
of the plane, the formula (ii) 

'= Vk V -^ becomes t=}iy/ ^=% \/-t-, 



/ "^ V A '* V A 



/ 



°'-''=i7=r (^5) 

Rule 15. — Given the length and the height of the inclined plane, to find 
the time in which a body would freely descend by gravitation. Divide the 
length by four times the square root of the height; the quotient is the time 
in seconds. 

For example, the length of the plane is 100 feet, and the height is 20 
feet, and the time is 

100 



^"47W =5-59 seconds, 



as was found before. 
Again, by inversion of the formula (15), 



= 4/\/ 



h , and then 



^=1^ (^^) 

Rule 16. — Given the length of the inclined plane, and the time of 
free descent by gravitation, to find the height through which the body 
descends. Divide the square of the length by the square of the time 
in seconds and by 16; the quotient is the length of the inclined plane. 

For example, the length of the plane is 100 feet, and the time of descent 
is 5.59 seconds; then the vertical height of the descent is 

h = 5 = 20 feet, the height. 

5.59^x16 

Average Velocity of a Moving Body Uniformly Accelerated 

OR Retarded. 

The average velocity of a moving body uniformly accelerated or retarded, 
during a given time or in a given space, is equal to half the sum of the 
initial and final velocities; and if the body begin from a state of rest or 
arrive at a state of rest, the average speed is half the final or initial velocity, 
as the case may be. Thus, in the example of a ball rolling, the initial 
speed or velocity is, in either case, 60 feet per second, and the terminal 
speed is nothing; the average speed is therefore the same, namely, one-half 
of that, or 30 feet per second 



MASS. — CENTRE OF GRAVITY. 287 

MASS. 

Weight IS not an essential property of a body; it is only the attraction of the 
earth exerted upon the body. Suppose the attractive force to be suspended, 
then the body would cease to have weight What would remain? Mass, 
or substance, simply. But, though weight is not mass, it is a direct measure 
of mass, in the same locality, or wherever the force of gravitation is the 
same, for double the mass has twice the weight Weight alone, however, 
is not sufficient as a universal measure of mass, since the weight of the 
same mass would vary according to the force of gravitation for different 
situations. The mass, therefore, varies inversely as the force of gravitation, 
when the weight remains the same. That force is measurable by the height 
through which a body falls in a given time, or by the velocity acquired at 
the end of that time, say, a second, expressed by g. In its most general 
form, then, the expression for the mass of a body comprises the weight 
directly and the force of gravitation inversely; thus 

^ = -T (17) 

in which m is the mass, w the weight, g the force of gravitation; that is to 
say, the mass of. a body is equal to the weight of the body divided by the 
force of gravity. Since the weight and the force of gravity vary in the 

same ratio, the mass — of a body is the same at all places. As the quan- 

tity of matter of the same body is also constant whatever place it occupies, 
the mass — gives an exact idea of the quantity of matter, and is a measure 
of it 

MECHANICAL CENTRES. 

There are four mechanical centres of force in bodies, namely, the centre 
of gravity, the centre of gyration, the centre of oscillation, and the centre 
of percussion. 

Centre of Gravity. 

The centre of gravity is the physical centre of a body, or of a system of 
bodies in rigid connection with each other, about which the gravitation of 
the several particles of the body is self-balanced, and at which it can be 
freely suspended or supported in any position in a state of rest. 

In various calculations, the whole weight or mass of a body is considered 
as placed at its centre of gravity. 

To find the centre of gravity of any plane figure mechanically: — Suspend 
the figure by any point near its edge, and mark on it the direction of a 
plumb-line hung from that point; then suspend it from some other point, 
and again mark the direction of the plumb-line in like manner. Then the 
centre of gravity of the surface will be at the point of intersection of the 
two marks of the plumb-line. 

The centre of gravity of parallel-sided objects may readily be found in 
this way. For instance, to find the centre of gravity of the arch of a bridge; 
draw the elevation upon paper to a scale, cut out the figure, and proceed 
v/ith it as above directed, in order to find the position of the centre of 



288 FUNDAMENTAL MECHANICAL PRINCIPLES. 

gravity in elevation for the model. In the actual arch, the centre of gravity 
will have the same relative position as in the paper model 

In regular figures or solids the centre of gravity is the same as their 
geometrical centres. Thus, the centre of gravity of a straight line, a 
parallelogram, a prism, a cylinder, a circle, the circumference of a circle, a 
ring, a sphere, and a regular polygon, is the geometrical centre of these 
figures and solids respectively. 

To find the centre of gravity of a triangle; draw a straight line from one 
of its angles to the middle of the opposite side; the centre of gravity will be 
in this line at a distance of two-thirds of its length from the angle. Or, 
draw a straight line from two of the angles to the middle of the opposite 
sides respectively; the point of intersection of the two lines will be the 
centre of gravity. 

For a trapezium, or irregular four-sided figure, draw the two diagonals, 
and find the centres of gravity of each of the four triangles thus formed ; 
then join each opposite pair of these centres of gravity. The joining lines 
will cut each other in the centre of gravity of the figure. 

For a cone and a pyramid, the centre of gravity is in the axis or centre 
line, at a distance of three-fourths of the length of the axis from the vertex, 
or one-fourth from the base. 

For an arc of a circle, the centre of gravity lies in the radius bisecting the 
arc, and the distance of it from the centre of the arc is found by multiplying 
the radius by the chord of the arc, and dividing by the length of the arc; the 
quotient is the distance of the centre of gravity from the centre of the circle. 

For a sector of a circle, the centre of gravity is two-thirds of the distance 
of that of an arc, from the centre of the circle. It is found independently 
by multiplying the radius by twice the chord of the arc, and dividing by 
three times the length of the arc ; the quotient is the distance of the centre 
of gravity from the centre of the circle. 

For a parabolic space, the centre of gravity is in the axis, or centre line, 
and its distance from the vertex is three-fifths of the centre line or axis. 

For a paraboloid, the centre of gravity is in the axis, at a distance from 
the vertex of two-thirds of the axis. 

For two bodies, fixed or suspended one at each end of a straight bar, the 
common centre of gravity is in the bar, at that point which divides the 
distance between their individual centres of gravity, in the inverse ratio 
of the weights respectively. For example, if two weights of 20 lbs. and 
10 lbs. be suspended on a bar at a distance of 9 feet apart between their 
centres of gravity, the common centre of gravity will divide the distance in 
the ratio of i to 2, being 3 feet from the heavier weight, and 6 feet from 
the lighter. In this example, the weight of the bar is neglected; but it may 
be allowed for according to the following direction. 

For more than two bodies connected in one system, the common centre 
of gravity may.be found by finding, in the first place, the common centre 
of gravity of two of them, and then finding that of these two jointly with 
a third, and so on to the last body in the group. 

Centre of Gyration. — Radius of Gyration. — Moment of Inertia. 

The centre of gyration, or revolution, is that point in a revolving body, or 
system of bodies, at a certain distance from tlie axis of motion, in which 
the whole of the matter in revolution may, as an equivalent condition, be 



CENTRE OF GYRATION. 289 

conceived to be concentrated, just as If a pound of platinum were substituted 
for a pound of revolving feathers, whilst the moment of inertia remains the 
same. The work accumulated in the body, of which the moment of 
inertia is a measure, remains in such a case the same, at the same angular 
velocity; and, as a necessary consequence, if the same accelerating force 
be applied to the body at the centre of gyration, as would actually be 
expended over the distributed matter of the body to communicate to it its 
angular velocity, the same angular velocity would be generated. 

The distance of the centre of gyration from the axis of motion is called 
the radius of g)rration ; and the moment of inertia is equal to the product 
of the square of the radius of gyration by the mass or weight of the body. 

The moment of inertia of a revolving body is found exactly by ascertain- 
ing the moments of inertia of every particle separately, and adding them 
together; or, approximately, by adding together the moments of the small 
parts arrived at by the subdivision of the body. 

Rule i. To find the moment of inertia of a revolving body. Divide 
the body into small parts of regular figure. Multiply the mass, or the 
weight, of each part by the square of the distance of its centre of gravity 
from the axis of revolution. The sum of the products is the moment of 
inertia of the body. 

Note. — ^The value of the moment of inertia obtained by this process 
will be more nearly exact, the smaller and more numerous the parts into 
which the body is divided. 

Rule 2. To find the length of the radius of gyration of a body about a 
given axis of revolution. Divide the moment of inertia of the body by its 
mass, or its weight, and find the square root of the quotient. The square 
root is the length of the radius of gyration; or 

'-/l- ; (') 

in which /// is the moment of inertia, and w is the weight of the body. 
Note, — When the parts into which the body is divided are equal, the 

radius of gyration may be determined by taking the mean of all the squares 

of the distances of the parts from the axis of revolution, and finding the 

square root of the mean square. 

The following are useful examples of the radius of gyration of bodies: — 
In a straight bar, or a thin rectangular plate, revolving about one of its 

ends, the radius of gyration is equal to the length of the rod, multiplied by 



^/ 



j^, or 0.5775. 



In a straight bar, or a thin rectangular plate, revolving about its centre, 
the radius of gyration is equal to half the length, multiplied by 



y 



y^y or 0.5775. 



The general expression for the radius of g}'ration in a straight bar revolving 
on any point of its length, is 






in which a and b are the lengths of the two parts of the bar; that is to say, 

19 



290 FUNDAMENTAL MECHANICAL PRINCIPLES. 

divide the sum of the cubes of the. two parts by three times the length of 
the bar, and extract the square root of the quotient. The root thus found 
is equal to the radius of gyration. 

In a circular plate, a solid wheel of uniform thickness, or a solid cylinder 
of any length, revolving on its axis, the radius of gyration is equal to the 
radius of the object, multiplied by 



y 



J4, or 0.7071. 



In a plane ring, like the rim of a fly-wheel, revolving on its axis, the radius 
of gyration is approximately equal to the square root of half the sum of 
the squares of the inside and outside radius of the rim. 

In a thin circular plate, put in motion round one of its diameters, 
the radius of gyration is equal to half the radius of the circle. 

For the circumference of a circle, revolving about a diameter, the radius 
of gyration is equal to the radius multiplied by 0.7071. 

In the circumference of a circle revolving about its ovm. axis, the radius 
of gyration is equal to the radius of the circle. 

In a solid sphere revolving about a diameter, the radius of gyration is 
equal to the radius multiplied by 



V 



V5, or 0.6324. 



In the surface of a sphere, or an insensibly thin spherical shell, the 
radius of gyration is equal to the radius multiplied by 



^/ 



^, or 0.8615. 



In a cone revolving about its axis, the radius of gyration is equal to the 
radius multiplied by 0.1783. 

Centre of Oscillation. 

The centre of oscillation of a body vibrating about a fixed axis or centre 
of suspension, by the action of gravity, is that point in which, if, as an 
equivalent condition, the whole matter of the vibrating body were concen- 
trated, the body would continue to vibrate in the same time. It is the 
resultant point of the whole vibrating energy, or of the action of gravity in 
causing oscillation. As the particles of the body further from the centre of 
suspension have greater velocity of vibration than those nearer to it, it is 
apparent that the centre of oscillation is more distant than the centre of 
gravity is from the axis of suspension, but it is situated in the centre line 
which passes from the axis through the centre of gravity. It differs also 
from the centre of gyration in this, that whilst the motion of oscillation is 
produced by the gravity of the body, that of gyration is caused by some 
other force acting at one place only. 

The radius of oscillation, or the distance of the centre of oscillation from 
the axis of suspension, is a third proportional to the distance of the centre 
of gravity from the axis of suspension and the radius of gyration. Hence 
the following rule for finding the radius of oscillation : — 



CENTRE OF OSCILLATION. — THE PENDULUM. 29I 

Rule 3. To find the radius of oscillation in a body vibrating on an 
axis. Square the radius of gyration of the body, and divide by the distance 
of the centre of gravity from the axis of suspension. The quotient is the 
radius of oscillation. Or, 

T» J- r n ^- radius^ of gyration. / ^ v 

Radius of oscillation = -j-. -z f^ 7-— p :— ( 3 ; 

distance of centre of gravity from axis. 

If the axis of suspension be in the vertex or uppermost point of a plane 
figure, and the motion be edgewise, then. 

In a right line, or straight rod, the radius of oscillation is two-thirds 
of the length of the rod. 

In an isosceles, or equal-sided triangle, it is three-fourths of the height 
of the triangle. 

In a circle it is five-eighths of the diameter. 

In a parabola it is five-sevenths of the height. 

But, if the oscillation of the plane figure be sidewise, then, 

In a circle suspended at the circumference, the radius of oscillation is 
three-fourths of the diameter. 

In a rectangle suspended by one of its angles, it is two-thirds of the 
diagonal. 

In a parabola suspended by the vertex, it is five-sevenths of the axis 
plus one-third of the parameter. 

In a parabola suspended by the middle of its base, it is four-sevenths 
of the axis plus half the parameter. 

In a sector of a circle suspended by the centre, it is three-fourths of the 
radius multiplied by the length of the arc, and divided by the length of the 
chord. 

In a cone it is four-fifths of the axis, plus the quotient obtained by 
dividing the square of the radius of the base by five times the axis. 

In a sphere it is two-fifths of the square of the radius divided by the sum 
of the radius and the length of the cord by which the sphere is suspended, 
plus the radius and the length of the cord. For example, in a sphere 
16 inches in diameter, suspended by a cord 25 inches long, the radius of 
oscillation is 

2x8^ 

-h 8 -I- 25 = 0.78 + 33 = 33.78 inches. 



5(8 + 25) 



or 0.78 inch below the centre of the sphere. 

It may be noted that the depression of the centre of oscillation below 
the centre of the sphere, namely, 0.78 inch, is signified in the first quantity 
in this equation. 

77^ Pendulum. 

A "simple pendulum" is the most elementary form of oscillating body, — 
consisting theoretically of a heavy particle attached to one end of a cord, 
or an inflexible rod, without weight, and caused to vibrate on an axis at 
the other end, or the centre of suspension. 

If an ordinary pendulum be inverted, so that the centre of oscillation shall 
become the centre of suspension, then the first centre of suspension will 
become the new centre of oscillation, and the pendulum will vibrate in the 




292 FUNDAMENTAL MECHANICAL PRINCIPLES. 

same time as before. This reciprocal action of the pendulum is a property 
of all pendulous bodies, and it is known as the reciprocity of the pendulum. 
The time of vibration of an ordinary pendulum depends on the angle or 
the arc of vibration, and is greater when the arc of vibration is greater, but 
in a very much smaller proportion; and if this arc do not exceed 4° or 5°, 
that is to say, from 2° to 2^° on each side of the vertical line, the time of 
vibration is sensibly the same, however the length of the arc may vary 
within that limit. This property of a pendulum, of equal times of vibration, 
is known as isochronism. 

To construct a pendulum such that the time of vibration shall be the 
same whatever the magnitude of the angle of vibration may be, it is neces- 
sary to cause the pendulum to vibrate, not in a circular arc, but in a 
cycloidal curve. For this object the pendulum is suspended by a flexible 
thread or rod, which oscillates between two cycloidal surfaces, on which it 
alternately laps and unlaps itself; these are generated by a circle of which 
the diameter is equal to half the length of the pendulum. By means of 
the circle o b. Fig. 98, for example, of which the diameter is half the length 

of the pendulum, describe the right and 
left cycloidal curves oca, go' a', on the 
horizontal line a a'; and draw the tangent 
c B c', touching the cycloids at the middle 
of their lengths. The half-lengths o c, o c', 
are equal to twice the diameter of the 
generating circle OB, and consequently 
equal to the length of the pendulum, which 
Fig. 98.— Cycloidal Pendulum. will vibrate in cqual times, on the centre 

of suspension o, between the entire half- 
lengths o c, o c', or in any shorter path. The curve c p c' thus described 
by the pendulum, is itself a cycloidal curve, and is a duplicate of the other 
cycloids. Though a cycloidal motion of the pendulum is necessary to render 
it isochronous for all angles of vibration, yet taking very small arcs of the 
cycloidal path on either side of the vertical line, they do not sensibly differ 
from the circular arcs which would be described by an ordinary pendulum 
of the same length (o p) swinging freely. Hence the reason that the ordinary 
I>endulum vibrates in equal times when its vibrations do not exceed 4° or 
5° in extent. 

The length of the pendulum vibrating seconds at the level of the sea 
in the latitude of London is 39.1393 inches, nearly a metre; at Paris 
it is 39.1279; at Edinburgh it is 39.1555 inches; at New York, 39.10153 
inches; at the equator it is 39.027 inches, and at the pole it is 39.197 
inches. Generally, if the force of gravity, or the length of the seconds 
pendulum at the equator be represented by i, the gravity, or the length of 
pendulum at other latitudes will be as follows : — 

Length of Seconds Pendulum, 

At the equator i. 00000 

„ 30'' latitude 1.00141 

„ 45 » 1.00283 

» 52 „ 1.00357 

,, 60 „ 1.00423 

„ 90 „ (the pole) 1.00567 



THE PENDULUM. 293 

According to these ratios, the force of gravity, and the length of the 
seconds pendulum, at the pole, are Viyeth greater than at the equator; there 
being a difference of length of between a fourth and a fifth of an inch. 

The following are the relations of the lengths of pendulums and the times 
of their vibrations, that is to say, of such as vibrate through equal angles, 
or of which the total angle of vibration does not exceed 4° or 5°: — 

The times of vibration of pendulums are proportional to the square root 
of the lengths of the pendulums. 

Conversely, the lengths oT pendulums are to each other as the squares of 
the times of one vibration, or inversely as the squares of the numbers of 
vibrations in a given time. 

The length of the seconds pendulum at London, 39.1393 inches, may 
be taken as a datum for calculation applicable to pendulums of different 
lengths, and to different times of vibration. 

Rule 4. To find the time of vibration of a pendulum of a given 
length. Divide the square root of the given length in inches by the 
square root of 39.1393, or 6.2561. The quotient is the time of a vibration 
in seconds. Or 

^ 391393 6.2561' 

in which / is the given length of pendulum in inches, and / the time of 
vibration in seconds. 

Rule 5. To find -the number of vibrations per second of a pendulum of 
given length. Divide 6.2561 by the square root of the length in inches. 
The quotient is the number of vibrations per second. 

For the number of vibrations per minute. Divide 375.366 by the 
square root of the length in inches. The quotient is the number of 
vibrations per minute. Or 

« = -^^^ (per second); ( 5 ) 

n = ^l^^ (^rmmute); (5) 

in which n is the number of vibrations. 

Rule 6. To find the length of a pendulum when the time of a 
vibration is given. Multiply the square of the time of one vibration in 
seconds by 39.1393. The product is the length of the pendulum in 
inches. Or 

/=/2x39.i393 (6) 

Rule 7. To find the length of a pendulum when the number of 
vibrations per second is given. Divide 39.1393 by the square of the num- 
ber of vibrations in a second. The quotient is the length of the pendulum 
in inches. 

When the number of vibrations per minute is given. Divide 140,900 
by the square of the number of vibrations in a minute. The quotient is 
the length of the pendulum in inches. Or 

/, 39-1393 /.) 

«^ (per second)' 

/_ 140,900 (7) 

n^ (per minute) 



294 FUNDAMENTAL MECHANICAL PRINCIFl.ES. 

A pendulum may be shorteneci and yet vibrate in the same time as 
before, by the action of a second weight fixed on the pendulum rod above 
the centre of suspension. Here the upper weight counteracts the lower, 
and there is only the balance of gravitating force due to the preponderance 
of the lower weight available for vibrating both masses. The mass being 
thus increased while the gravitating force is diminished, a longer time is 
required for each vibration when the length of pendulum remains unaltered, 
or the pendulum may be shortened so that the time of the vibrations con- 
tinues the same. By varying the height of the upper weight above, the 
centre of suspension, and thus varying the level of the common centre of 
gravity, the period of vibration is varied in proportion. 

Rule 8. To find the weight of the upper bob of a compound pendulum 
necessary to vibrate seconds, when the weight of the lower bob is given, 
and the respective distances of the bobs from the centre of suspension. 
Multiply the distance in inches of the lower bob from the centre of suspen- 
sion by 39.1393, and from the product subtract the square of that distance 
(i); again, multiply the distance in inches of the upper bob from the 
centre of suspension by 39.1393, and add the square of that distance 
(2); multiply the lower weight by the remainder (i), and divide by the 
sum (2). The quotient is the weight of the upper bob. Or 

^ = w 99-i393xD)- D^ (8) 

(39-1393 >"^ + <^' 

in which D and d are the respective distances of the lower and upper bobs 
from the centre of suspension, and W, a/, their respective weights. 

Thus, by means of a second bob, pendulums of small dimensions 
may be made to vibrate as slowly as may be desired. The metronome, 
an instrument for marking the time of music, is constructed on this 
principle, the upper weight being slid and adjusted on a graduated rod 
to measure fast or slow movements. 

The Centre of Percussion. 

If a blow is struck by an oscillating or revolving body moving about a 
fixed centre, the percussive action is the same as if the whole mass of 
the body were concentrated at the centre of oscillation. That is to say, 
the centre of percussion is identical with the centre of oscillation, and 
its position is found by the same rules as for the centre of oscillation. 
If an external body is so struck that the mean line of resistance passes 
through the centre of percussion, then the whole force of percussion is 
transmitted directly to the external body; on the contrary, if the revolving 
body be struck at the centre of percussion, the motion of the revolving 
body will be absolutely destroyed, so that the body shall not incline either 
way, just as if every particle separately had been struck. 

CENTRAL FORCES. 

When a body revolves on an axis, every particle moves in a circle of 
revolution, but would, if freed, move off in»a straight line, forming a tangent 
to the circle. The force required to prevent the body or partiple flying 
from the centre is called cctiiripetal force, and the tendency to fly from 
the centre is centrifugal force. These forces are equal and opposite — 
examples of action and reaction — and are classed as central forces. 



CENTRAL FORCES. 295 

Centrifugal force varies as the square of the speed of revolution. 

It varies as the radius of the circle of revolution. 

It varies as the mass or the weight of the revolving body. 

Let c be the centrifugal force, w the weight of the revolving body, 

r the radius of revolution or gyration, m the mass of the body = -, in which 

^ = 32.2 or gravity; and v the linear or circumferential velocity; then 

m v^ w v^ 

c— - = 

r 32.2 r 

That is to say, the centrifugal force of a revolving body is equal to the 
weight of the body multiplied by the square of the linear velocity, divided 
by 32.2 times the radius of gyration. 

If the height due to the velocity be substituted for the velocity in the 

above equation, the height // being equal to - — , then 

64.4 

2WV^ 2 w h 

€=- = , 

64.4 r r 

and 

c \ w \ \ 2 h \ K 

That is to say, the centrifugal force is to the weight of the body as twice the 
height due to- the velocity is to the radius of gyration. 

From the first equation the following rules for revolving bodies are 
deduced, for finding one of the four elements when the other threie are 
given: — namely, the centrifugal force, the radius of gyration, the linear 
velocity, and the weight. 

Rule i. For the centrifugal force. Multiply the weight by the square 
of the speed, and divide by 32.2 times the radius of gyration. The quotient 
is the centrifiigal force. Or 

c^ (i) 

32.2 r 

Rule 2. For the linear velocity. Multiply the centrifugal force by the 
radius of gyration, and by 32.2, and divide by the weight; and find the 
square root of the quotient. The root is the velocity. Or 



/x2.2cr / V 

^ w 



Rule 3. For the weight Multiply the centrifugal force by the radius of 
gyration, and by 32.2, and divide by the square of the velocity. The 
quotient is the weight. Or 

32.2 c r / V 

'^ = ^-:jr- : ; (3) 

Rule 4. For the radius of gyration. Multiply the weight by the square 
of the velocity, and divide by the centrifugal force, and by 32.2. The 
quotient is the radius of gyration. Or 



w iP' 



r-= (4) 

32.2 c 



296 FUNDAMENTAL MECHANICAL PRINCIPLES. 

Note, — ^When the velocity is expressed as angular velocity, in revolutions 
per unit of time, it is to be reduced to linear or circumferential velocity by 
multiplying it by the radius of gyration and by 6.28; or 

f = 6.28 if r, 

in which 7/ is the angular velocity. 

By substitution and reduction in equation (i), the following equation in 
terms of the angular velocity is arrived at : — 

0.8165 c=wr7/^, /j) 

from which is found 

'=5176i = '-"S«"-^' (6) 

That is to say, the centrifugal force is equal to the weight multiplied by the 
radius of gyration and by the square of the angular velocity, and by 1.225. 

MECHANICAL ELEMENTS. 

The function of mechanism is to receive, concentrate, diffuse, and apply 
power to overcome resistance. The combinations of mechanism are num- 
berless; but the primary elements are only two, namely, the lever and the 
inclined plane. By the lever, power is transmitted by circular or angular 
action; that is to say, by action about an axis; by the inclined plane, it is 
transmitted by rectilineal action. The principle of the lever is the basis of 
the pulley and the wheel and axle; that of the inclined plane is the basis of 
the wedge and the screw. 

For the present, frictional resistance and the weight of the mechanism 
are not considered; the terminal resistance is called the weight; and the 
elemental mechanisms are to be treated as in a state of equilibrium, in 
which the power exactly balances the weight without actual movement. 
The action, or work done, will be subsequently treated. 

The Lever. 

The elementary lever is an inflexible straight bar, turning on an axis or 
fixed point, called the fulcrum; the force being transmitted by angular 

motion about the fulcrum, from the 
point where the power is applied to the 
point where the weight is raised, or 
other resistance overcome. There are 
three varieties of the lever, according 
as the fulcrum, the weight, or the 
power is placed between the other two, 
but the action is, in every case, re- 
Pjg. 99-— Lever. duciblc to that of three parallel forces 

in equilibrium (page 275). 
First. The power is applied at one end ^, of the lever ab c. Fig. 99, and 
transmitted through the fulcrum, ^, to the weight at the other end c. The 
moments of the power and the weight about the fulcrum are equal, or 

power y^ab- weight y>b c. 

That is, the product of the power by its distance from the fulcrum is equal 




THE LEVER. 



297 



to the product of the weight by its distance from the fulctum. Conse- 
quently 

power : weight : : b c \ ab^ 

that is, the power and the weight are to each other inversely as their 
respective distances from the fulcrum. 

The ratio of the length of the power end of the lever to the length of the 
weight end is called the leverage of the power. The respective lengths, 
Fig. 99, being 7 feet and i foot, the leverage is 7 to i, or 7, 

The three varieties of the lever are 
grouped together in Figs. 100, 10 1, and 1 

102. In each case, the lever is supposed I _, I 

to be 8 feet long and divided into feet. J ^ f \ -, — p 1  . ^. 
The leverage, in the first, is 7 to i, or 7; ^^ | "S 

in the second, 8 to i, or 8; in the third, 'f^ 

yi to I, or ^ : showing that, in the first Q I A 

case, the power balances seven times its yr ^ 

own amount; in the second case, eigh 
times its amount; in the third case, only 



Fig. xoo. — Lever, ist kind. 



^ + 

R 



T ' '  -r- 



Ji 



a 




II'''' 



IT 



Fig. loi. — Lever, ad kind. 



Fig. loa. — Lever, 3d kind. 



one-eighth of itself, because it is nearer to the fulcrum than the weight. 

In each case the moments of the power and the weight about the fulcrum 
are equal, for, in each case. 



Pxdf ^ = Wx3^. 



{a) 



The pressures exerted at the extremities of the lever act in the same 
direction, and the sum of them is equal and opposite to the intermediate 
pressure, whether it be that of the fulcrum, the weight, or the power ( — ). 
From this the pressure on the fulcrum may be found. If it be in the 
middle, the pressure is equal to the sum of the power and the weight, that 
is, 60 + 420 = 480 lbs. in the example above ; if at one end, it is equal to 
the difference of them, that is, it is 480 — 60 = 420 lbs. when the weight is 
in the middle, and it is 60-7^^ = 52)^ lbs. when the power is in the 
middle. 

From the equation for the equality of moments, 

orPxL =Wx/, {b) 

in which L and / are the respective distances of the power and the weight 
from the fulcrum, rules may be formed for finding any one of the four 
quantities, when the other three are given. 

Rule i. To find the power. Multiply the weight by its distance from 
the fulcrum, and divide by the distance of the power from the fulcrum. 
The quotient is the power. 



298 FUNDAMENTAL MECHANICAL PRINCIPLES. 

Or, divide the weight by the leverage. The quotient is the power. Or 

Rule 2. To find the weight Multiply the power by its distance firom 
the fulcrum, and divide by the distance of the weight from the fulcrum. 
The quotient is the weight. 

Or, multiply the power by the leverage. The product is the weight Or 

w=y- (2) 

Rule 3. To find the distance of the power from the fulcrum. Multiply 
the weight by its distance from the fulcrum, and divide by the power. The 
quotient is the distance of the power from the fulcrum. Or 

^ = 17 (3> 

Rule 4. To find the distance of the weight from the fulcrum. Multiply 
the power by its distance from the fulcrum, and divide by the weight The 
quotient is the distance of the weight from the fulcrum. Or 

9 If the weight of the lever be included in such calculations, its influence 
is the same as if its whole weight or its mass were collected at its centre of 
gravity. Thus, if the lever of the first kind, Fig. 100, weighs 30 lbs., and its 
centre of gravity be at the middle of its length, the weight of the lever 
co-operates with the power, at a mean distance of 3 feet from the fulcrum. 
By equality of moments 

(P X 7) X (30 X 3) = W X I = 420 lbs. X I, 
and P X 7 = 420 - 90 = 330 lbs.; 

therefore P, the power at the end of the lever required to balance the 




Fig. Z03. — Inclined Lever. 




Fig. xo4.^Inclined Lever. 



weight, is only 3304-7 = 47.1 lbs. in co-operation with the weight of the 

lever, as compared with 60 lbs., without reckoning the aid from this source. 

When the lever is inclined to the direction of the forces, as in Fig. 103, 



THE LEVER. 



299 



equilibrium, or the equality of moments, may nevertheless be maintained. 
Drawing the horizontal line a' b d through the fulcrum, to meet the ver- 
ticals through the power and the weight at a! and ^, the moments of the 
power and the weight are to be estimated on the horizontal lengths ci b, b c'\ 
and 

the moment V %a! b- the moment W xb i/. 

The equality of moments may be proved in another way. Let the 
power and the weight be resolved, in order to find the pressures on the 
ends of the lever, at right angles to it, and thus to arrive at the moments 
as estimated on the actual length of the lever. Let the verticals through 
the ends of the lever, a m and cn^ Fig. 104, represent the power and the 
weight respectively, and draw a Y and c W perpendicular to the lever, and 
/// P' and n W parallel to if, completing the triangles a m P', c n W. Then 
a P' and c W are the components of the power and the weight respectively 
tending to turn the lever; and, it may be added, they bear the same ratio 
to each other as the power and the weight Consequently, if these com- 
ponents be multiplied by the respective lengths of the lever, the products 
will be the moments of the components, and the moments will be equal; or 

the moment ^ P' x df ^ = the moment cW xb c. 

These two methods of analyzing and finding the moments of the forces 
acting on an inclined lever — one, combining a reduced length of lever witjj 
the whole power and weight; the other, combining the whole length of 
lever with a reduced power and weight — lead to one conclusion, that a 
lever, if balanced in one position, is balanced in every other position, 
when the forces continue to act in parallel lines. 





■® 



O 



Fig. 105.— Bent Lever. 



Fig. X06. — Bent Lever. 



The conditions of equilibrium in a bent lever may be defined sinriilarly. 
Let the lever a b c. Fig. 105, be bent at the fulcrum b; draw the horizontal 
line of b d, then the moments of the power and the weight are reckoned 
on the lines a' b, b c*, and they are equal to each other; or 



300 



FUNDAMENTAL MECHANICAL PRINCIPLES. 



»' «• 



Again, let the forces acting on a lever, whether straight or bent, be 
otherwise than vertical or parallel. When the arms of the lever are at 
right angles, and the power and the weight applied at right angles to the 
arms, as in Fig. io6, the moments are reckoned directly on the arms, ab,bc^ 
as in a straight lever; and 

the moment P x « ^ ^ the moment W x ^ r. 

The thrust, or pressure on the fulcrum, is in this case less than the sum of 
the power and the weight; and it may be determined by constructing a 

parallelogram upon the two arms of the 
lever, the arms representing inversely the 
respective forces. That is, a b represents 
the magnitude and direction of the weight 
W, and b c th®se of the power P. The 
diagonal b y, of the parallelogram repre- 
sents the magnitude and direction of the 
third force acting at the fulcrum to oppose 

O^ ^ the other two and maintain equilibrium. 

When the same lever is tilted into an 
IT oblique position, the power continuing to 

Fig. 107.— Bent Lever. act horizontally on the lever, Fig. 107, 

draw the vertical b' d through the end c of 
the lever, and produce the power line ap \.o meet it at }/, Complete the 
parallelogram a! 1/ (f b; then the sides a* b and b d zi^ the perpendiculars 
to the directions to the power and weight, on which the moments are 
reckoned, so that 

the moment P x a' ^ = the moment W x ^ ^. 
The diagonal ^ ^ is the resultant force at the fulcrum. 





^ 

^ 



6 




Fig. X08. — Bent I.,ever. 



Fl?. T09. — Serpentine Lever. 



If the power do not act horizontally, but in some other direction, a /, 
Fig. 108, produce the power-line pa and draw ba^ perpendicular to it. 



THE LEVER. 3OI 

Draw b (f 2& before ; then the moments are reckoned on the perpendiculars 
b (fy b c\ and, as before, 

To find the resultant force at the fulcrum. On the fulcrum ^ as a centre 
describe arcs of circles with the radii b a' and b (f, and draw b a^y b (f 
respectively parallel to the directions of the weight and the power, to cut 
the arcs at (f and c'^. Complete the parallelogram, and the diagonal b b^ 
represents in magnitude and direction the resultant force at the fulcrum. 

In this solution the power and the weight are assumed to act exactly, 
or sensibly, in the same plane. 

Again, in the serpentine lever a b c^ Fig. 109, supposed to be a pump- 
handle, the power P is applied obliquely in the direction a P. Produce 
P a and W c, and draw tlie perpendiculars b af^b <f from the fulcrum for 
the lengths of the- moments, then 

Pxrt'^ = Wx^^. 

Construct the parallelogram, as in the foregoing figure, and the diagonal 
b b" represents the resultant force at the 
fulcrum. ^/ 

By similar treatment the action of the . :'" ^^^^^"^. 

forces in levers of the second and third : 




kinds may be analyzed. The lever of the ®'~'-^,"yf^^ Av 

second kind, a c by Fig. 1 10, in an oblique \ \^^^^ C-/ 

position, is acted on horizontally by the : * -P 

power and the weight at a and c; draw /'^ 

the vertical b d a!y then b c' and b of are \^ 
the distances at which the forces act from W 

the fulcrum, or the lengths of the mo- Fig. no.— Lever of the ad kind. 

ments, and 

and the horizontal resultant force at the fulcrum is equal to the difference 
of the weight and the power. 

If more than two forces be applied to a lever in a state of equilibrium, 
the sum of the moments tending to turn tlie lever in one direction is equal 
to the sum of those tending in the opposite direction. 

If two or more levers are connected consecutively one to the other, so 
that they act as one system, with the power and the weight at the extremi- 
ties, then, in equilibrium, the ratio of the power to the weight is the product 
of the separate inverse ratios of all the levers. For example, in a connected 
series of three levers, having each their arms in the ratio of 2 to i, the 
combined inverse ratio is found by multiplying 2 by 2 and by 2 ; thus 

first lever 2 to i ratio, 

second lever 2 to i ratio, 

third lever 2 to i ratio, 

compound ratio 8 to i. 

That is; the power is to the weight as i to 8. 



302 



FUNDAMENTAL MECHANICAL PRINCIPLES. 



The Pulley. 

The pulley is a wheel over which a cord, or chain, or band is passed, in 
order to transmit the force applied to the cord in another direction. It is 
equivalent to a continuous series of levers, with equal arms on one fulcrum 
or axis, and affords a continuous circular motion instead of the intermittent 
circular motion of one lever. The weight W, Fig. iii, is sustained by the 
power P, by means of a cord passed over the pulley A, in fixed supports, 
and the centre line abc represents the element of the lever, from the ends 
of which the power and the weight may be conceived to depend, turning 
on the fulcrum b. By equality of moments, V y.ab = Vf y.bc; and the 
arms or radii a b, b c being equaJ, the power is equal to the weight, and the 
counter-pressure at the fulcrum is equal to twice the weight. 

When the power and weight act in directions #t an angle with each other, 
as in Fig. 112, the acting radii ab, be, representing the element of a bent 



^^V1lh^Vll»^^^vftf^^^1^l 




s>x-.v^ 7-vA.-*»a>vvvv •■ .,.^.-^^ 




•vr 




Fig. xxL—Pullcy. 



Fig. 112.— Pulley. 



Fig. 1x3. —Pulley. 



lever, are lines drawn from the centre perpendicular to the directions of the 
power and weight The power is equal to the weight, but the counter- 
pressure on the fulcrum is less than twice the weight, and is represented by 

the diagonal b V of the parallelogram formed by the 
radii bc^, bd, drawn from the fulcrum parallel to 
the directions of the power and the weight respec- 
tively. 

Another construction for the parallelogram of 
forces in the action of the pulley is obtained by 
producing the directions of the power and the weight 
beyond the pulley. Fig. 113, intersecting each other 
at y, then forming the parallelogram, and drawing 
the diagonal b' if 2& the resultant pressure on the 
fulcrum. 

Thus the single fixed pulley acts like a lever of the 
first kind, and simply changes the direction of force, 
without modifying the intensity of the power. 

But the pulley may be employed as a lever of the 
second kind by suspending the weight to the axis 
of the pulley, and fixing one end. of the cord to a point as a fulcrum 
point. Thus, in Fig. 114, the weight W is suspended from the axis c. 




Fig. 114. — Movable Pulley, 
as a lever of the 3d kind. 



THE PULLEY. $0$ 

the cord is fixed to the point i', and the power P acts through the diameter 
acd, in which ^ is the fulcrum. By equality of moments, 

that is, the product of the power by the diameter of the pulley is equal to 
the product of the weight by the radius of the pulley, 
and the leverage being as 2 to 1, the power is only 
half the weight. 

In acting as a lever of the third kind, the power is 
applied to the axis a. Fig. 115, one end of the cord 
b«ing fixed at ^, and the weight attached at the other 
end, c. In this case, by equality of moments the 
product of the power by Jhe radius of the pulley is 
equal to that of the weight by the diameter, and the 
leverage being as i to z, the power is twice the weight 

These demonstrations with respect to movable 
pulleys only apply to cases of parallel cords; that is 
to say, when the direction of the power is parallel to J,ie»trofihe dkini" 
that of the weight. If, on the contrary, they be inclined ' 

to each other, as in Fig. 116, in which the weight is suspended by the 
axis, the power becomes greater than half the weight, as is shown by the 
parallelogram of which the diagonal c'tf represents the weight, and the 
sides e'y, i (f, the pull on the fulcrum, and the power exerted to sustain 
the weight Each of these sides is greater than half the diagonal. 





Fig. ii«.-MovaMe PuBey. Fir. iij.-FullcyBladci. 

ComMrta/ions of PuUeys. — Fast and Loose Pulleys. — In these last two 
applications of the pulley, it becomes movable when in action, and by com- 
bining two or more movable pulleys on the same or different axles in one 
block, with one cord, the gain of power may be increased in the same pro- 
portion. The movable block A, Fig. 117, carrying the weight, is used 



304 FUNDAMENTAL MECHANICAL PRINCIPLES, 

with a fixed counterpart E, the rope is attached by one end to the fixed 

block, and is passed over the movable and fixed pulleys, from one to Che 

other in succession, the power being applied to the other end, as at P. 

'I'his system is known as &st and loose puliey-b locks. 

The fixed end of the rope is sometimes attached to the movable block. 

Rule i. To find the power necessary to balance a weight or resistance 

by means of a system of fast and loose pulleys. Divide the weight by the 

is carried; that is, the number of ropes which 

alock. The quotient is the power required to 

2 rope is attached to the fixed block, the num- 

Ti the loose block is twice the number of mov- 

may be found by dividing the weight by twice 

:ys. • 

 is attached to the movable block, the divisor 

imber of movable pulleys plus i. 

ber of movable pulleys; if the fixed end of the 

block, 

I: (.) 

ipe be attached to the movable block, 

^ (..) 

ight or resistance that will be balanced by a 
system of fast and loose pulleys. Multiply the 
r by the number of ropes proceeding from the 
Lble block. The product is the required weight 
■, when the rope is attached to the fixed block, 
ply the power by twice the number of movable 
ys. 

■, when the rope is attached to the movable 
:, multiply the power by twice the number of 
Lble pulleys plus i. 
, in the first case, 

W-2»P; (2) 

t: second case, 

w.(»«+.)P M 

;ain, a combination may be formed of a num- 
f movable pulleys, as in Fig. 118, each of which, 
C, is suspended by a cord, with one end fixed 
e roof and the other end fixed to the axis of 
ext pulley. The weight W is hung to the axis 
delivers half the weight to the second pulley B, 
veight hanging to it, or one-fourth of the first 
ey C; from which only one-eighth of the first 
de or neutral pulley D lo the power P. In 



THE PULLEY. 



30s 



general the divisor for the power is 2", or the //th power of 2, n being the 
number of movable pulleys. 

Rule 3. To find the power necessary to balance a weight by means of a 
system of separate movable pulleys, with separate cords consecutively con- 
nected as above described. Divide the weight by that power of 2 of which 
the index is the number of movable pulleys. The cjuotient is the power or 
force required to balance the weight 

Or, divide and subdivide the weight successively by 2 as many times as 
there are movable pulleys to find the power required. Or 



P = 



W 



(3) 



Rule 4. To find the weight that can be balanced by a given power, by 
means of a system of separate movable pulleys as above described. Mul- 
tiply the power by that power of 2 of which the index is the number of 
movable pulleys. The product is the weight required. 

Or, multiply the power successively by 2 as many times as there are 
pulleys. Or 

W=PX2'' (4) 

Note. — It is necessary that the cords should be parallel to each other, 
as in the illustration, in order that the above rules, 3 and 4, may apply. 



Wheel and Axle. 

The wheel and axle may be likened to a couple of pulleys of different 
diameters united together on one axis, of which the larger, a, Fig. 1 19, is the 
wheel, and the smaller, c, the axle, with a common ful- 
crum, b', the centre line abc representing the elements of 
a lever. The weight W on the axle at c is balanced *by 
the power P, on the wheel at a. The moments are equal, 
or 

Pxfl!^ = Wx^r; 

and the power is to the weight inversely as their distances 
from the centre; or 

V :^Y :\ be : ab. 




Fig. 119.— Wheel 
ana Axle. 



If a crank handle be substituted for the wheel, making 
a windlass, the radius of the crank is substituted for that 
of the wheel in estimating the ratio of the forces. 

Of the four elements, namely, the radius of the wheel or crank, the radius 
of the axle or roller, the power, and the weight, if three be given, the fourth 
can be found as follows, putting R and r for the respective radii. 

Rule i. To find the power. Multiply the weight by the radius of the 
axle, and divide by the radius of the wheel. The quotient is the power. 
Or 



WxJ 



(i) 



Rule 2. To find the weight Multiply the power by the radius of the 

20 



3o6 



FUNDAMENTAL MECHANICAL PRINCIPLES. 



wheel, and divide by the radius of the axle. The quotient is the weight 
Or 



W = Px — 
r 



(») 



Rule 3. To find the radius of the wheel. Multiply the weight by the 
radius of the axle, and divide by the power. The quotient is the radius of 
the wheel. Or 

R=^^ (3) 



Rule 4. To find the radius of the axle. Multiply the power by the 
radius of the wheel,* and divide by the weight. The quotient is the radius 
of the axle. Or 

PR / V 
(4) 



r- 



W 



Note, — The diameters of the wheel and the axle or roller may be 
employed in the calculations instead of the radii. 

Inclined Plane. 

The inclined plane is a slope, or a flat surface inclined to the horizon, on 
which weights may be raised. By such substitution of a sloping path for 
a direct vertical line of ascent, a given weight can be raised by a power 
which is less than the weight itself. 

There are three elements of calculation in the inclined plane: — the plane 
itself, A B, Fig. 120; the base, or horizontal length, AC; and the height or 

vertical rise B C ; together forming a right- 
angled triangle. The weight W is to be 
raised through a height equal to C B, and 
for that object is drawn up the slope from 
A to B. It is partly supported during the 

O^^^^v^ (^^)^^ ascent, and it is in virtue of this degree of 

^'*"^^\i^7/ support given to the weight that such a 

^dFC^ "dead pull" as that of a direct vertical 
lift is avoided, and that it can be raised 
by a power much less than its o\mi weight. 
Let the weight W be kept at rest on the 
incline by the power P, acting in the line b P', parallel to the plane. Draw 
the vertical line ^^ to represent the weight; also bV perpendicular to the 
plane, and complete the parallelogram V c. Then the vertical weight ba 
is equivalent to b b\ which is the measure of support given by the plane to 
the weight, and b c, which is the force of gravity tending to draw the weight 
down the plane. The power required to maintain the weight in equilibrium 
is represented by this force be. Thus, the power and the weight are in 
the ratio of beta b a. 

Since the triangle of forces abe is similar to the triangle of the incline 
A B C, the latter may be substituted for the former in determining the 
relative magnitude of the forces, and 

P : W :: ^^ : fl^ :: BC : AB, 




Fig. X30.--Incluied Plane. 



THE INCLINED PLANE. 



307 



that is, the power, acting parallel to the inclined plane, is to the weight, as 
the height of the plane to its length. Then, by equality of moments, 

PxAB = WxBC, 
or P X length of inclined plane = W x height of inclined plane (a) 

For example, take the length of the inclined plane, 24 feet; the height, 
2 feet; and the weight to be raised, 360 lbs. The power required to 
balance the weight is equal to 360 x 2 h- 24 = 30 lbs. 

Again, the base, A C, of the inclined plane, represents the element of 
the pressure of the weight on the inclined plane. 

It is thus seen that the sides of the triangle formed by an inclined plane, 
its base, and its height, are respectively proportional as follows : — 

The inclined plane to the weight at rest on the plane. 

The base to the pressure of the weight on the plane. 

The height to the power acting parallel to the plane. 

When the power acts in a direction parallel to the base, as in Fig. 121, in 
which the power P supports the weight 
W in the direction d V\ parallel to the 
base; draw the vertical da to represent 
the weight, and the line d^ perpen- 
dicular to the incline, and complete the 
parallelogram d' c. The weight If a, de- 
composed, is equivalent to ^ ^', the per- 
pendicular to the incline, representing 
the pressure of the weight upon the 
plane, and d c, the force of traction, or 
the power P. Here the pressure ^ ^ on 
the plane is greater than the v/eight, and 
the power ^ ^ is greater than when the line of traction is parallel to the 
incline. 

The triangles adc, ABC, being similar, the ratios of the power and the 
weight are as follows : — 

F :W :: dc : ad :: BC : AC; {3) 

that is, they are to each other as the height of the plane to its base; and 
the inclined plane, the base, and the height, are respectively proportional 
as follows: — 

The inclined plane to the pressure of the weight on the plane. 
The base to the weight at rest on the plane. 

The height to the power acting parallel to- the base. 

If the power be applied in any direction above that which is parallel to 
the incline, though the pressure of the weight on the plane will be less than 
the weight itself, yet, as in the previous case, the power is greater than is 
necessary when it acts in a direction parallel to the plane. Thus, in 
Fig. 122, in which the power P acts at a divergent angle in the direction 
d P', draw the vertical da, the perpendicular dd\ to the plane, and the 
extension of the power line to c, and complete the parallelogram. Then, 
the weight is represented by d a, the pressure on the incline by d V, and the 
power by a ^ or ^ ^. 




Fig. lax. — Inclined Plane. 



3o8 



FUNDAMENTAL MECHANICAL PRINCIPLES. 




Fig. laa.— Inclined Plane. 



For comparison, the parallelogram that would represent the relative 
forces arising from a power acting parallel to the plane, is added on the 
figure in dotted lines extending to the angles b" and (f. It shows that the 

pressure on the plane is greater 
than when the power is di- 
vergent, but that the power 
is less. 

It follows that the longer 
the inclined plane, when the 
height is the same, the less is 
the power required to balance 
the weight; in fact, the power 
simply varies in the inverse 
ratio of the length of the 
plane. 

If two inclines, A B and 
B D, of unequal lengths and 
the same height, be united 
back to back on the line BC, 
then two weights, W and W, 
on the respective planes, connected by a cord over a pulley at the summit 
B, will balance each other, when they are in the ratio of the lengths of the 
planes on which they rest That is, 

W : W : : A B : B D. 

From the formula ( a ), rules may be formed for finding one of the following 
four elements when the other three are given, namely, the length of the 

inclined plane, the height of 
it, the weight, and the power 
to balance the weight when 
acting in a direction parallel 
to the incline. 

Rule i. To find the power. 
Multiply the weight by the 
height of the plane, and divide 
by the length. The quotient 
is the power. 

Rule 2. To find the weight Multiply the power by the length of the 
plane, and divide by the height. The quotient is the weight 

Rule 3. To find the height of the inclined plane. Multiply the power 
by the length, and divide by the weight The quotient is the height 

Rule 4. To find the length of the inclined plane. Multiply the weight 
by the height of the plane> and divide by the power. The quotient is the 
length. 

Identity of the Inclined Plane and the Lever, 

Though the inclined plane is distinguished from the lever in the mode of 
operation, inasmuch as there is no motion about a mechanical centre, as in 
the lever, yet the conditions of equilibrium on the inclined plane may be 
established on the principle of the lever. Suppose a round weight W kept 
at rest on the incline A B by a power P parallel to the incline, passing 




Fig. 123. — Double Inclined Plane. 



LEVERAGE ON THE INCLINED PLANE. 



309 



through the centre a. Draw ab perpendicular to the incline; the point b 

is the point of contact of the weight with the incline. Draw the vertical 

line a d, and the perpendicular ^ ^ to it. Then the lines ab, be form a bent 

lever a be, of which b is the fulcrum, and 

ab,be the arms. The weight acts at the 

extremity e of the short arm, and the power 

at the extremity a of the long arm; and the 

power and the weight are to each other 

inversely as the relative arms of the lever, 

ab,be. Now, as abe and A B C are similar 

triangles, the arms a b, be axe to each other 

as the length and the height A B, B C, of 

the incline, and 




'P : W : : b e : a b : : B C : A B: 



Fig. 124. — Leverage ofh an Inclined Plane*. 



that is, the power is to the weight as the height of 
length, which is the proportion already established 

The ratio of the length of an inclined plane to 
the leverage of the plane, and the products of the 
the plane, and of the weight into the height of the 
moments of the power and the weight. 

Suppose, again, that the power is applied at P, 
a P, passed round and over the weight parallel 



the inclined plane to its 
( « ) page 307). 
its height may be called 
power into the length of 
plane, may represent the 

Fig. 125, through a cord 
to the incline; then the 




C A 

Fig. 125. — Leverage on an Inclined Plane. 




Fig. I a6.— Wedge. 



diameter of the weight a b becomes the long arm of the lever a be, through 
which the power acts, being double the length of the arm a b, Fig. 1 24, 
where the power is applied at the centre of the weight. By thus doubling 
the leverage, the power is halved, and the ratio of the power to the weight 
is as half the height of the plane to its length. 

In this case there is the action of a movable pulley combined with an 
inclined plane; the rolling weight moved by a cord lapped round it, repre- 
senting a movable pulley with the weight attached to the axle. Thus the 
leverage of the power on the inclined plane can be doubled. 

The Wedge. 

The wedge is a pair of inclined planes united by their bases, or " back to 
back," as A B C B , Fig. 1 26. Whereas inclined planes are fixed, wedges are 
moved, and in the direction of the centre line C A, against a resistance 
equally acted on by both planes of the wedge. The function of the wedge 



310 FUNDAMENTAL MECHANICAL PRINCIPLES. 

is to separate two bodies by force, or divide into two a single body. In 
some cases the w^edge is moved by blows, as in splitting timber; in others 
it is moved by pressure. The action by simple pressure is now to be con- 
sidered. 

The pressure P is applied to a wedge at the head B B' at right angles to 
the surface, and the resistance or "weight" to be overcome is opposed to 
the wedge and acts at right angles to the faces A B, A B', at the middle 
points of which, a, a, it is supposed, to be concentrated. Whilst the wedge 
and the power move through a space equal to the length of the wedge A C, 
the weight is moved or overcome through a space equal to the breadth of 
the wedge B B'; and, as the power is to the weight inversely as the spaces 
described, they are to each other directly as the breadth to the length of 
the wedge. That is, 

P : W : : B B' : A C, 

and the product of the power by the length of the wedge is equal to the 
product of the weight by the breadth of the wedge; or 

PxAC = WxBB; 

or P X length = W X breadth of wedge {c) 

By the aid of the parallelogram the same conclusions are arrived at 
Thus, in Fig. 126, produce the directions of the two resistances, W«, \N a, to 
meet in the middle of the wedge at by complete the parallelogram, and draw 
the diagonals aca and bb\ The diagonal b b' is the resultant of the two 
forces ab,ab, and represents the pressure on the head of the wedge. Again, 
in the triangle a be, whilst a b represents, in magnitude and direction, the 
perpendicular pressure of the resistance on the wedge, a c, which is perpen- 
dicular to the centre line of the wedge, represents, in magnitude and 
direction, the force applied in overcoming the resistance. The ratio of the 
power to the weight is therefore as bb' to a c. And, as the triangle abb' is 
similar to the triangle ABB', 

P : W :: ^^ : ^z^ :: BB' : AC; 

that is, the power is to the weight as the breadth of the wedge to its length. 

From the formula ( c ), the following rules for wedges acting under pres- 
sure, as distinct from impact, are deduced : — 

Rule i. To find the weight or transverse resistance. Multiply the 
power by the length of the wedge, and divide by the breadth of the head. 
The quotient is the weight. 

Rule 2. To find the power. Multiply the weight or transverse resistance 
by the breadth of the head, and divide by the length of the wedge. The 
quotient is the power. 

Rule 3. To find the length of the wedge. Multiply the weight by the 
breadth of the wedge, and divide by the power. The quotient is the 
length of the wedge. 

Rule 4. To find the breadth of the wedge. Multiply the power by the 
length of the wedge, and divide by the weight. The quotient is the breadth 
of the wedge. 

Note, — I. The length of the wedge is taken as the distance from the 
head to the point of intersection of the sides. 



THE SCREW. 



311 



2. The power may be applied at the point of the wedge by drawing, 
instead of at the head by pressing. 

3. The power may be applied in a direction parallel to one of the sides 
of the wedge, and the relation of the power to the weight may be found by 
construction, in the same manner as for the inclined plane, when the power 
is applied in a direction parallel to the base. See proportion ( ^ ), page 307. 

The Screw. 

The screw is an inclined plane lapped round a cylinder. Take, for 
example, an inclined plane ABC, Fig. 127, and bend it into a circular form, 
resting on its base, Fig. 128, so that the ends meet The incline may be 




Fig. 127. 

continued winding upwards round the same axis, and thus winding or 
helical inclined planes of any required length and height may be con- 
structed. The helix thus arrived at being placed upon a solid cylinder, 
and the dead parts of the helix removed, the product is an ordinary screw. 
The inclined fillet is the " thread " of the screw, and the screw is called 
"external." But the thread may also be applied 
within a hollow cylinder, and then it is " internal," 
such as an ordinary " nut " is. 

The distance of two consecutive coils apart, 
measured from centre to centre, or from upper side 
to upper side, — ^literally the height of the inclined 
plane, — ^for one revolution, is Qie "pitch" of the 
screw. 

The effect of a screw is estimated in terms of the 
pitch and the radius of the handle employed to turn either it or the nut, 
one on the other; and the leverage of the power is the ratio of the circum- 
ference of the circle described by the power end of the handle to the pitch. 
The radius is to be measured to the central point where the power is 
applied. 

The circumference being equal to the radius multiplied by twice 3. 14 16, 
or 6.28, 




Fig. ia8. 



P : W :: / : /-X6.28, 
in which / is the pitch and r the radius; also 

6.28 Pr=Wx/; 



{d) 



that is, 6.28 times the product of the power by the radius of the handle is 
equal to the product of the weight by the pitch. Whence the following 
rules relative to the power of a screw, for finding any one of those four 
quantities when the other three are given : — 

Rule i. To find the power. Multiply the weight by the pitch, and 



312 FUNDAMENTAL MECHANICAL PRINCIPLES. 

divide by the radius of the handle and by 6.28. The quotient is the 
power. Or 

P = ^^ (X) 

6.28 r ^^ 

Rule 2. To find the weight. Multiply the power by the radius and by 
6.28, and divide by the pitch. The quotient is the weight. Or 

^^6,28^r ^^j 

Rule 3. To find the pitch. Multiply the power by the radius of the 
handle and by 6.28, and divide by the weight. The quotient is the pitch. 
Oi 

^ 6.28 Pr , V 

/=— w~ : ^^) 

Rule 4. To find the radial length of the handle. Multiply the weight 
by the pitch, and divide by the power and by 6.28. The quotient is the 
length of the handle. Or 

"j5^p- <^) 

JYofe. — When the power is applied through a wheel fixed to the screw, 
the acting diameter of the wheel may be substituted for the radius in the 
above rules and formulas, and the constant becomes 3.14. 

Similarly, should the power-wheel be fixed to the nut so as to turn 
it upon the screw, instead of the screw within the nut, the same sub- 
stitutions may be made. 

WORK. 

Work consists of the sustained exertion of pressure through space. 

The English unit of work is one foot-pound; that is, a pressure of one 
pound exerted through a space of one foot. 

The French unit of work is one kilogrammetre; that is, a pressure of one 
kilogramme exerted through a space of one metre. 

One kilogrammetre is equal to 7.233 foot-pounds. 

In the performance of work by means of mechanism, the work done 
upon the weight is equal to the work done by the power. This prin- 
ciple of the equality of work is deducible from the principle of the 
equality of moments, and is expressed generally by the equation 

PxH = Wx>4, (a) 

in which H is the height or space moved through by the power, and 
// the height or space moved through by the weight at the same time. 
It signifies that the product of the power by the space through which it has 
acted is equal to the product of the weight by the space through which 
it has acted. 
Again, 

P : W : : /i : H, 

signifying that the power is to the weight inversely as the respective heights 
or spaces moved through by them in the same time. 



WORK. — WORK WITH THE MECHANICAL ELEMENTS. 313 

Work done with the Lever. 

On the principle of the equality of moments, the power and the weight 
in the lever, neglecting frictional resistance, are to each other inversely as 
the lengths of the arms upon which they act, that is, of their radii of 
motion; and inversely as the arcs or spaces passed through or described by 
the ends of the arms. If the weighted lever, Fig. 99, page 296, be moved 
by the power, the power descends through an arc at a, and the weight is 
raised through an arc at c. These arcs may be taken as the heights moved 
through, and are proportional to the lengths of the respective arms, ab^b c. 
In this example, these are as 7 to i, and if the power descend 7 inches the 
weight is raised only i inch; but the weight raised is seven times the power 
applied, and "what is gained in power is lost in speed," or, more correctly, 
in space moved through. The equality of work thus developed from the 
equality of moments is thus expressed 

power X arc a — weight x arc c («) 

To show this arithmetically, let the weight be raised through i foot; then, 
with a leverage of 7 to i, the power descends 7 feet, and taking it, as before, 
at 60 lbs., the weight it raises will be 60 lbs. x 7 = 420 lbs., and the equation 
of work is 

60 lbs. X 7 feet = 420 lbs. x i foot, 
(or 420 foot-pounds) (or 420 foot-pounds). 

Again, 

power : weight : : arc c : arc a^ 

expressing the principle of virtual velocities, the relative velocities being 
indicated by the arcs «, c. 

Work done with the Pulley. 

In using the single fixed pulley. Fig. iii, page 302, the power is equal to 
the weight, and the spaces through which they move in the same time are 
equal. 

With the movable pulley, Fig. 114, the weight is suspended at the axle, 
and in raising the weight i foot, the power at the circumference, with a 
leverage of 2, passes through 2 feet and is only half the weight If 
P and W be 20 lbs. and 40 lbs. respectively, the equality of work is thus 
expressed — 

(P) 20 lbs. X 2 feet = (W) 40 lbs. x i foot = 40 foot-pounds; 

and by means of this pulley a weight double the power is raised half the 
height through which the power is applied. 

Conversely, when the weight is suspended at the circumference of the 
movable pulley, Fig. 115, and the power applied at the axle, the leverage 
is J^ ; the power is therefore double the weight, and moves through i foot 
whilst the weight moves through 2 feet. Thus 

(P) 40 lbs. X I foot = (W) 20 lbs. X 2 feet = 40 foot-pounds. 

In a system of fast and loose pulley blocks, Fig. 117, page 303, the 
power being equal to the weight divided by the number of ropes, then, by 



314 FUNDAMENTAL MECHANICAL PRINCIPLES. 

equality of work, the space through which the power is moved is equal to 
the height through which the weight is raised, multiplied by the number of 
ropes. Suppose that there are three movable pulleys and six ropes; if the 
weight, izo lbs., be raised i foot) each rope is shortened i foot and the 
power is moved 6 feet And 

(P) 20 lbs. X 6 feet = (W) 120 lbs. x i foot = 120 foot-pounds. 

Work done with the Wheel and Axle. 

'heel, Fig. iig, page 305, makes one revolution, the axle also 
The power descends or traverses a space equal to the cir- 
thewheel = 2 (ai) x 3.i4i6,whilst the weight is raised through 

to the circumference of the axle = 2 (i^:) x 3.1416. If the 
wheel be r foot 6 inches, and that of the axle 3 inches, the 
s are 9.42 feet and 1.57 feet, being as 6 to i; and the power 
t, conversely, are as 1 to 6. If the power be 20 lbs., then 
;P) 20 lbs. X g.42 feet = {W) rao lbs. x 1.57 feet. 

(188.4 foot-pounds) (188.4 foot-pounds). 

Work done with the Inclined Plane. 

is raised in opposition to gravity, and the work done on it is 
the product of the weight into the vertical height of the 

;. The work done by the power is enpressed by the product 
into the length of the'plane, These two products express 

es of work, and 

Px/=Wx^, 

nated at (a), page 307, to express equality of moments. 

e, the length of the plane is 24 feet and the height 2 feetj 

1,20 lbs., the power 10 lbs. Then, the work done in rising 

the whole of the incline is 240 lbs., thus 

(P) 10 lbs. X 24 feet = (W) 1 20 lbs. x 2 feet 
(240 foot-pounds) {240 foot-pounds). 

lere supposed to be applied in a direction parallel to the plane. 
a direction at an angle to the plane, as in Fig. 122, page 308, 
lolved into its components, parallel and perpendicular to the 
the line ^ <;' parallel to the incline; then the power applied, 
;nt to the force actually expended & 1!, and to the pressure 
)n c c". The consumption of power is expressed by the pro- 
irallel equivalent, b c, into the length of tiie plane. Taking, 
LS above, the weight, 120 lbs., and the active power, 10 lbs., 
ly the parallel force b tf; then the amount of the horizontal 
)ower applied, b c, is found by proportion, thus 

AC:ABi:i/:if; 

arallel and horizontal forces are to each other as the base to 
the incline. 



WORK IN MOVING BODIES. 315 

Work done with the Wedge. 

Supposing the wedge driven by a constant pressure through a distance 
equal to its length, the work done by the power is expressed by the power 
into the length, and the work done on the weight is expressed by the pro- 
duct of the weight into the breadth of the wedge. By equality of work, 

PxL = WxB, 

as before stated, in expressing equality of moments. 

If the wedge be driven for only a part of its length, the work done by 
the power is in the proportion of the part of the length driven; and the 
work done on the weight is similarly in the proportion of the part of the 
breadth by which the resisting surfaces are separated. 

Work done with the Screw. 

In one revolution of the screw, the weight is raised through a height 
equal to the pitch of the thread, whilst the power acts through the circum- 
ference of the circle described by the point at which it is applied to a lever. 
The products of the power and the weight by the spaces (^escribed by 
them are equal, or 

Px6.28r = Wx/, 
as before stated (page 311) to express equality of moments. 

Work done by Gravity. 

The work done by gravity on a falling body is equal to the weight of the 
body multiplied by the height through which it falls. 

Work accumulated in Moving Bodies. 

The quantity of work stored in a body in motion is the same as that 
which would be accumulated in it by gravity if it fell from such a height as 
would be sufficient to give it the same velocity; in short, from the height 
due to the velocity. (See Graviit, page 2^^),^ The accumulated work 
expressed in foot-pounds, is equal to the height so found in feet, multiplied 
by the weight of the body in pounds. The height due to the velocity is 
equal to the square of the velocity divided by 64.4, and the work and the 
velocity may be found directly from each other, according to the following 
rules: — 

Rule i. Given the weight and velocity of a moving body, to find the 
work accumulated in it. Multiply the weight in pounds by the square of 
the velocity in feet per second, and divide by 64.4. The quotient is the 
accumulated work in foot-pounds. 

Or, putting U for the work, v for the velocity, and 7v for the weight, 

U = 4^ (I) 

64.4 

Or, secondly: — Multiply the weight in pounds by the height in feet due 
to the velocity. The product is the accumulated work in foot-pounds. Or, 
putting A for the height, 

U = «/x// ( i^) 



3l6 FUNDAMENTAL MECHANICAL PRINCIPLES. 

Work done by Percussive Force. 

If a wedge be driven by blows or strokes of a hammer or other heavy 
mass, the effect of the percussive force is measured by the quantity of work 
accumulated in the striking body. This work is calculated by the preceding 
rules, from the weight of the body and the velocity with which the blow is 
delivered, or directly from the height of the fall, if gravity be the motive 
power. 

The useful work done through the wedge is equal to the work delivered 
upon the wedge, supposing that there is no elastic or vibrating reaction 
from the blow, just as if the work had been delivered by a constant pres- 
sure equal to the weight of the striking body, exerted through a space equal 
to the height of the fall, or the height due to its final velocity. 

Of course, in order to give effect to the constant pressure on the wedge, 
now imagined to be brought into action, the pressure would require to be 
applied to the resisting medium through some combination of the mechanical 
elements. 

But where elastic action intervenes, a portion of the work delivered is 
uselessly absorbed in elastically straining the resisting body; and the elastic 
action may be, in some situations, so excessive as to absorb the whole of 
the work delivered. In this case, there would not be any useful work done. 

These remarks, applied to the action of a blow on a wedge, are applicable 
equally to the action of a blow of the monkey of a pile-driver upon a pile. 
If there be no elastic action, the work delivered being the product of the 
weight of the monkey by the height* of its fall, is equal to the work done in 
sinking the pile; that is, to the product of the frictional and other resistance 
to its descent by the depth through which it descends for one blow of the 
monkey. 

Supposing that the pile rests upon and is absolutely resisted by a hard 
unyielding obstacle, the work done becomes wholly useless, and consists of 
elastic or vibrating action ; or it may be that the head of the pile is split 
open. 



HEAT. 



THERMOMETERS. 

The action of Thermometers is based on the change of volume to which 
bodies are subject with a change of temperature, and they serve, as their 
name implies, to measure temperature. Thermometers are filled with air, 
water, or mercury. Mercurial thermometers are the most convenient, because 
the most compact. They consist of a stem or tube of glass, formed with a 
bulbous expansion at the foot to contain the mercury, which expands into 
the tube. The stem being uniform in bore, and the apparent expansion of 
mercury in the tube being equal for equal increments of temperature, it 
follows that if the scale be graduated with, equal intervals, these will indi- 
cate equal increments of temperature. A sufficient quantity of mercury 
having been introduced, it is boiled to expel air and moisture, and the tube 
is hermetically sealed. The freezing and the boiling points on the scale 
are then determined respectively by immersing the thermometer in melting 
ice and afterwards in the steam of water boiling under the mean atmospheric 
pressure, 14.7 lbs. per square inch, and marking the two heights of the 
column of mercury in the tube. The interval between these two points is 
divided into 180 degrees for Fahrenheit's scale, or 100 degrees for the 
Centigrade scale, and degrees of the same interval are continued above and 
below the standard points as far as may be necessary. It is to be noted 
that any inequalities in the bore of the glass must be allowed for by an 
adaptation of the lengths of the graduations. The rate of expansion of 
mercury is not strictly constant, but increases ¥dth the temperature, though, 
as already referred to, this irregularity is more or less nearly compensated 
by the varying rates of expansion of glass. 

In the Fahrenheit Thermometer, used in Britain and America, the number 
0° on the scale corresponds to the greatest degree of cold that could be 
artificially produced when the thermometer was originally introduced. 32° 
("the freezing-point") corresponds to the temperature of melting ice, and 
212® to the temperature of pure boiling water — in both cases under the 
ordinary atmospheric pressure of 14.7 lbs. per square inch. Each division 
of the thermometer represents i® Fahrenheit, and between 32** and 212° 
there are i8o^ 

In the Centigrade Thermometer, used in France and in most other 
countries in Europe, o** corresponds to melting ice, and 100® to boiling 
water. From the freezing to the boiling point there are 100°. 

In the R^umur Thermometer, used in Russia, Sweden, Turkey, and 
Egypt, o** corresponds to melting ice, and 80° to boiling water. From the 
freezing to the boiling point there are 80°. 



3 1 8 HEAT. 

Each degree Fahrenheit is | of a degree Centigrade, and y of a degree 
Reaumur, and the relations between the temperatures indicated by the 
different thermometers are as follows : — 

C. = I (F. -32). R. = I (F. -32). C. = f R. 

G. being the temperature in degrees Centigrade. 
R. do. do. Reaumur. 

F. do. do. Fahrenheit. 

That is to say, that Centigrade temperatures are converted into Fahrenheit 
temperatures by multiplying the former by 9 and dividing by 5, and adding 
32° to the quotient; and conversely, Fahrenheit temperatures are converted 
into Centigrade by deducting 32°, and taking |ths of the remainder. 

Reaumur degrees are multiplied by | to convert them into the equivalent 
Centigrade degrees; conversely, |ths of the number of Centigrade degrees 
give their equivalent in Reaumur degrees. 

Fahrenheit is converted into Reaumur by deducting 32° and taking |ths 
of the remainder, and Reaumur into Fahrenheit by multiplying by f , and 
adding 32® to the product 

Tables No. 104, 105 contain equivalent temperatures in degrees Centigrade 
for given degrees Fahrenheit, from 0° F., or zero on the Falirenheit scale, to 
608° F. ; and conversely, the temperature in degrees Fahrenheit correspond- 
ing to degrees Centigrade, from 0° C, or zero on the Centigrade scale, to 
320° C. 



EQUIVALENT TEMPERATURES. 



319 



Table Na 104. — Equivalent Temperatures by the Fahrenheit 

AND Centigrade Thermometers. 



Degrees 


Degrees 


Degrees 


Degrees 




Degrees 


Degrees 


Degrees 


Fahr. 


Centigrade. 


Fahr. 


Centigrade. 


Fahr. 


Centigrade. 


Fahr. 


Centigrade. 





-17.78 


+ 38 


+ 3.34 


+ 76 


+ 24.45 


+ 114 


+ 45.56 


+ I 


17.23 


39 


3.90 


77 


25.00 


1^5 


46.11 


2 


16.67 


40 


4.45 


78 


25.56 


116 


46.67 


3 


16.II 


41 


5.00 


79 


26.12 


117 


47.23 


4 


15.56 


42 


5.56 


80 


26.67 


118 


47.78 


5 


15.00 


43 


6. 1 1 


81 

I 


27.23 


119 


48.34 


6 


14.45 


44 


6.67 


82 


27.78 


120 


48.90 


7 


13.90 


45 


7.23 


^3 


28.34 


121 


49.45 


8 


13.34 


46 


7.78 


84 


28.89 


122 


50.00 


9 


12.78 


47 


8.34 


85 


29.45 


123 


50.56 


10 


12.23 


48 


8.89 


86 


30.00 


124 


5 I.I I 


II 


11.67 


49 


9-45 


87 


30.55 


125 


51-67 


12 


II. II 


so 


10.00 


88 


31. II 


126 


52.23 


13 


10.56 


51 


10.56 


89 


31.67 


127 


52.78 


14 


10.00 


52 


II. II 


90 


32.22 


128 


53-34 


15 


9.45 


S3 


11.67 


91 


32.78 


129 


53.90 


16 ' 


8.89 


54 


12.23 


92 


33-33 


130 


54.45 


17 


8.34 


55 


12.78 


93 


33.89 


131 


55-00 


18 


7.78 


56 


13.34 


94 


34.45 


132 


55-56 


19 


7.23 


57 


13.90 


95 


35.00 


133 


56.11 


20 


6.67 


58 


14.45 


96 


35.56 


134 


56.67 


21 


6. 1 1 


59 


15.00 


97 


36.11 


135 


57.23 


22 


5-56 


60 


15.56 


98 


36.67 


136 


57.78 


23 


5.00 


61 


16.11 


99 


37.23 


137 


58.34 


24 


4.45 


63 


16.67 


100 


37.78 


138 


58.90 


25 


3.90 


63 


17.23 


lOI 


38.34 


139 


59.45 


26 


3.34 


64 


17.78 


102 


38.90 


140 


60.00 


27 


2.78 


65 


18.34 


103 


39.45 


141 


60.56 


28 


2.23 


66 


18.89 


104 


40.00 


142 


61. II 


29 


1.67 


67 


19.45 


105 


40.56 


143 


61.67 


30 


I. II 


68 


20.00 


106 


41. II 


144 


62.23 


31 


0.56 


69 


20.56 j 


107 


41.67 


145 


62.78 


32 


0.00 


70 


21. II 


108 


42.23 


146 


63.34 


33 


+ 0.56 


71 


21.67 


109 


42.78 


147 


63.90 


34 


I. II 


72 


22.23 


no 


43.34 


148 


64.45 


35 


1.67 


73 


22.78 , 


III 


43.90 


149 


65.00 


36 


2.23 


74 


23-34 


112 


44.45 


150 


65.56 


37 


2.78 


75 


23.90 


"3 


45.00 


151 


66.11 



HEAT. 
Table No. 104 {continued). 



■^fX" 


Demes 
C«l«rade. 


^^ 


Deerea 


^S^." 


Cenugrade. 


f5^ 


Dcgr«s 
Cenugndt. 




66.67 


+ 193 


+ 89.45 


+ 234 


■HI 12.23 


+ 275 


+ 135-00 


67-^3 


194 


90.00 


235 


112.78 


376 


I35.56 


67.78 


195 


90.56 


236 


"3-34 


277 


136" 


68.34 


196 


91. II 


237 


113.90 


278 


136.67 


6S.90 


197 


91.67 


238 


"4-45 


279 


137-23 


69-45 


198 


92-23 


239 


115.00 


380 


137-78 


70.00 


"99 


93.78 


240 


115.56 


381 


138.34 


70.56 


300 


93-34 


241 


116.11 


383 


138.90 


71.11 


201 


93-90 


242 


116.67 


2S3 


139.45 


71.67 


Z02 


94-45 


243 


117-23 


284 


140.00 


73.33 


203 


95-O0 


344 


117.78 


285 


140.56 


73.78 


304 


95.56 


245 


118.34 


386 


141. II 


73-34 


205 


96.11 


246 


118.90 


287 


141.67 


73-9° 


206 


96.27 


247 


"9-45 


288 


142.23 


74-45 


207 


97-23 


248 


120.00 


389 


142.78 


75-00 


208 


97.78 


249 


120.56 


390 


143.34 


75-56 


209 


98-34 


250 


121.11 


291 


133-90 


76.1. 




98.90 


25" 


131.67 


292 


144.45 


76.67 


211 


99-4S 


252 


132.23 


293 


I45-00 


77-^3 


312 


100.00 


253 


122.78 


394 


145-56 


77.78 


213 


100.S6 


254 


'23.34 


295 


146. 1. 


78.34 


214 


lOI.II 




123-90 


296 


146.67 


78.90 


315 


101.67 


256 


124-45 


397 


147.23 


79-45 


zi6 


102.23 


257 


125.00 


398 


147.78 




217 


102.78 


258 


125.56 


299 


148.34 


80.56 


3l8 


103-34 


259 


136.11 


300 


148.90 


81. II 


219 


103.90 


260 


126.67 


301 


149-45 


81.67 




104-45 


261 


127-23 


302 


150.00 


83.23 


221 


105.00 


262 


137.78 


303 


150-56 


83.78 


232 


105.56 


363 


128.34 


304 


151.11 


8334 


223 


106. 1 1 


264 


128.90 


305 


151.67 


83.90 


224 


106.67 


26s 


129.45 


306 


152.23 


84.4s 


22s 


107.23 


266 


130.00 


307 


152.78 


85.00 


336 


107.78 


267 


130.56 


308 


153.34 


85.56 


227 


108.83 


268 


131-11 


309 


'53.90 


86.11 


328 


108.90 


369 


131-67 


310 


154-45 


86.67 


229 


109.45 


270 


132.23 


3" 


155.00 


8723 


230 




371 


132-78 


312 


J5S-56 


87.78 


231 


"a?6 


272 


133-34 


313 




88.34 


23a 


iii.ii 


273 


133-90 


314 


Ise'eS 




88.90 


"33 


111.67 


274 


134-45 


3«S 


157-23 



EQUIVALENT TEMPERATURES. 



321 



Table No. 104 {continued). 







Fahrenheit and Centigrade. 






D^re«s 


De^ees 


Degrees 


Decrees 


Degrees 


De^ees 


Degrees 
Fahr. 


Degrees 


Fahr, 


Centigrade. 


Fahr. 


Centigrade. 


Fahr. 


Cenugrade. 


Centigrade. 


+ 316 


+ 157.78 


+ 357 


+ 180.56 


+ 398 


+ 203.34 


+ 439 


+ 226.11 


317 


15^-34 


358 


181. II 


399 


203.90 


440 


226.67 


318 


158.90 


359 


181.67 


400 


204.45 


441 


227.23 


319 


159.45 


360 


182.23 


401 


205.00 


442 


227.78 


320 


160.00 


361 


182.78 


402 


205.56 


443 


228.34 


321 


160.56 


362 


183.34 


403 


206.11 


444 


228.90 


322 


161. II 


363 


183.90 


404 


206.67 


445 


229.45 


323 


161.67 


364 


184.45 


405 


207.23 


446 


230.00 


324 


162.23 


365 


185.00 


406 


207.78 


447 


230.56 


325 


162.78 


366 


185.56 


407 


208.34 


448 


231. II 


326 


163.34 


367 


186. 1 1 


408 


208.90 


449 


231.67 


327 


163.90 


368 


186.67 


409 


209.45 


45^ 


232.23 


328 


164.45 


369 


187.23 


410 


210.00 


451 


232.78 


329 


165.00 


370 


187.78 


411 


210.56 


452 


233.34 


330 


165.56 


371 


188.34 


412 


211. II 


453 


233.90 


331 


166.II 


372 


188.90 


413 


211.67 


454 


234.45 


332 


166.67 


373 


189.45 


414 


212.23 


455 


235.00 


333 


167.23 


'374 


190.00 


415 


212.78 


456 


235.56 


334 


167.78 


375 


190.56 


416 


213.34 


457 


236.11 


335 


168.34 


376 


191. II 


417 


213.90 


458 


236.67 


336 


168.90 


377 


191.67 


418 


214.45 


459 


237.23 


337 


169.45 


378 


192.23 


419 


215.00 


460 


237.78 


338 


170.00 


379 


192.78 


420 


215.56 


461 


238.34 


339 


170.56 


380 


193.34 


421 


2l6.II 


462 


238.90 


340 


171. II 


381 


193.90 


422 


216.67 


463 


239.45 


341 


171.67 


382 


194.45 


423 


217.23 


464 


240.00 


342 


172.23 


383 


195.00 


424 


217.78 


465 


240.56 


343 


172.78* 


384 


195.56 


425 


218.34 


466 


241. II 


344 


173.34 


385 


I96.II 


426 


218.90 


467 


241.67 


345 


173.90 


386 


196.67 


427 


219.45 


468 


242.23 


346 


174.45 


387 


197.23 


428 


220.00 


469 


242.78 


347 


175.00 


388 


197.78 


429 


220.56 


470 


243.34 


348 


• 175.56 


389 


198.34 


430 


221. II 


471 


243.90 


349 


1 76. 1 1 


390 


198.90 


431 


221.67 


472 


244.45 


350 


176.67 


391 


199.45 


432 


222.23 


473 


245.00 


351 


177.23 


392 


200.00 


433 


222.78 


474 


245.56 


352 


177.78 


393 


200.56 


434 


223.34 


475 


246.11 


353 


178.34 


394 


201. II 


435 


223.90 


476 


246.67 


354 


178.90 


395 


201.67 


436 


224.45 


477 


247.23 


355 


179.45 


396 


202.23 


437 


225.00 


478 


247.78 


356 

> 


180.00 


397 


202.78 


438 


225.56 


479 


248.34 














21 





Table No. 104 (cotUitnud). 
'ahrenmeit and Ckntigrao 





Ccnli^ijdc 


DegMS 


Centigrade, 


■iX" 


Degrees 
Cemigrade. 


'^^ 


C™.«™ic, 




+ 248.90 


+ 5"3 


+ 267.23 


+ 546 


+ 285.56 


+ 579 


+ 303-90 




249-45 


SI4 


267-78 


547 


286.11 


5B0 


304-45 




250.00 


515 


268-34 


548 


286.67 


58' 


305.00 




250.56 


516 


268-90 


549 


287.23 


582 


305-5'' 




25I-" 


5"7 


269.45 


55° 


287.78 


583 


306.11 




2S,.67 


5>8 


270.00 


55' 


288.34 


584 


306.67 




252.23 


5>9 


270.56 


552 


288.90 


585 


307-23 




252.78 


520 


271.11 


553 


289.45 


586 


307.78 




253.34 


521 


271.67 


554 


290.00 


S87 


308.34 




253-90 


522 


272.23 


555 


290.56 


588 


308.90 




254-45 


523 


272.78 


556 


291. II 


589 


309-45 




255.00 


524 


273-34 


557 


291.67 


590 


310.00 




255.56 


525 


27390 


558 


292.23 


591 


310.56 




256.,. 


526 


27445 


559 


292.78 


592 


311. II 




256.67 


527 


275.00 


560 


29334 


593 


311.67 




257-23 


528 


275-5' 


561 


293.90 


594 


312.23 




257.78 


529 


276-1. 


562 


294.45 


595 


312.78 




258-34 


530 


276.67 


563 


295.00 


596 


313-34 




25890 


531 


277-23 


564 


295-56 


597 


313-90 




25945 


532 


277-78 


565 


296.11 


598 


314-45 




260-00 


533 


278.34 


566 


296.67 


599 


315-00 




260-56 


534 


278.90 


567 


297.23 


600 


315-56 




261-11 


535 


279.45 


568 


297.78 


601 


316.11 




261.67 


536 


280.00 


569 


298.34 


602 


316.67 




262.23 


537 


280.56 


570 


298.90 


603 


317-23 




262.78 


538 


281.11 


571 


299-45 


604 


31778 




263.34 


539 


281.67 


572 


300.00 


605 


3T8.34 




263.90 


540 


282.23 


573 


300.56 


 606 


318.90 




264.45 


541 


282.78 


574 


301. 11 


607 


319-45 




265.00 


542 


283-34 


575 


301.67 


608 


320.00 




265-56 


543 


283-90 


576 


302.23 








266.11 


544 


284-45 


577 


302.78 








366.67 


545 


285.00 


578 


303.34 







EQUIVALENT TEMPERATURES. 



323 



Table No. 105. — Equivalent Temperatures by the Centigrade and 

Fahrenheit Thermometers. 



Degrees 


Degrees 


Degrees 


Degrees 
Fahr. 


Degrees 


Degrees 


Degrees 


Degrees 


Cent. 


Fahr. 


CenL 


Cent. 


Fahr. 


Cent. 


Fahr. 


-20 


- 4.0 


+ 21 


+ 69.8 


+ 62 


+ 143-6 


+ 103 


+ 217.4 


19 


2.2 


22 


71.6 


63 


145.4 


104 


219.2 


18 


0.4 


23 


73-4 


64 


147.2 


105 


22l!o 


17 


+ 1.4 


24 


75.2 


65 


149.0 


106 


222.8 


16 


3.2 


^5 


77.0 


66 


150.8 


107 


224.6 


15 


5.0 


26 


78.8 


67 


152.6 


108 


226.4 


H 


6.8 


27 


80.6 


68 


154.4 


109 


228.2 


13 


8.6 


28 


82.4 


69 


156.2 


no 


230.0 


12 


10.4 


29 


84.2 


70 


158.0 


III 


231.8 


II 


12.2 


30 


86.0 


71 


159.8 


112 


233.6 


10 


14.0 


31 


87.8 


72 


161.6 


"3 


235-4 


9 


15.8 


32 


89.6 


73 


163.4 


114 


237.2 


8 


17.6 


33 


91.4 


74 


165.2 


"5 


239.0 


7 


19.4 


34 


93-2 


75 


167.0 


116 


240.8 


6 


21.2 


35 


95.0 


76 


168.8 


117 


242.6 


5 


23.0 


36 


96.8 


77 


170.6 


118 


2444 


4 


24.8 


^l 


98.6 


78 


172.4 


119 


246.2 


3 


26.6 


38 


100.4 


79 


174.2 


120 


248.0 


2 


28.4 


39 


102.2 


80 


176.0 


121 


249.8 


I 


30.2 


40 


104.0 


81 


177.8 


122 


251.6 





32.0 


41 


105.8 


82 


179.6 


123 


253.4 


+ I 


33.8 


42 


107.6 


83 


181.4 


124 


255.2 


2 


35.6 


43 


109.4 


84 


183.2 


125 


257.0 


3 


37.4 


44 


III. 2 


85 


185.0 


126 


258.8 


4 


39-2 


45 


1 130 


86 


186.8 


127 


260.6 


5 


41.0 


46 


1 14.8 


87 


188.6 


128 


262.4 


6 


42.8 


47 


1 16.6 


88 


190.4 


129 


264.2 


7 


44.6 


48 


1 18.4 


89 


192.2 


130 


266.0 


8 


46.4 


49 


120.2 


90 


194.0 


131 


267.8 


9 


48.2 


50 


122.0 


91. 


195.8 


132 


269.6 


10 


50.0 


51 


123.8 


92 


197.6 


133 


271.4 


II 


51.8 


52 


125.6 


93 


199.4 


134 


273.2 


12 


53.6 


53 


127.4 


94 


201.2 


135 


275.0 


13 


55-4 


54 


129.2 


95 


203.0 


136 


276.8 


14 


57.2 


55 


1310 


96 


204.8 


137 


278.6 


15 


59.0 


56 


132.8 


97 


206.6 


138 


280.4 


16 


60.8 


57 


134.6 


98 


* 

208.4 


139 


282.2 


17 


62.6 


58 


136.4 


99 


210.2 


140 


284.0 


18 


64.4 


59 


138.2 


100 


212.0 


141 


285.8 


19 


66.2 


60 


140.0 


lOI 


213.8 


142 


287.6 


20 


68.0 


61 


141.8 


102 


215.6 


143 


289.4 



324 



HEAT. 



Table No. 105 {continued). 
Centigrade and Fahrenheit. 



Depves 


Jjcgtccs 


Degrees 


Degrees 




Degrees. 


DcgFMS 




Cent. 


Fahr. 


CenL 


Fahr. 


C«lL 


Fahr. 


Cent. 


Fahr. 


+ 144 


+ 291.2 


+ 189 


+ 372.2 


1 
+ 234 


+ 453.2 


+ 279 


+ 534-2 


H5 


293.0 


190 


. 374.0 


235 


455.0 


280 


536.0 


146 


294.8 


191 


375-8 


236 


456.8 


281 


537.8 


147 


296.6 


192 


377.6 


237 


458.6 


282 


539.6 


148 


298.4 


193 


379.4 


238 


460.4 


283 


541.4 


149 


300.2 


194 


381.2 


239 


462.2 


284 


543.2 


150 


302.0 


195 


383.0 


240 


464.0 


285 


545.0 


151 


303-8 


196 


384.8 


. 241 


465.8 


286 


546.8 


152 


305.6 


'97 


386.6 


242 


467.6 


287 


548.6 


153 


307.4 


198 


388.4 


243 


469.4 


288 

1 


5504 


154 


309.2 


199 


390.2 


244 


471.2 


i 289 


552.2 


'55 


31 10 


200 


392.0 


245 


473.0 


I 290 


554.0 


156 


312.8 


201 


393.8 


246 


474.8 


291 


555.8 


157 


314.6 


202 


395-6 


247 


476.6 


292 


557.6 


158 


316.4 


203 


397.4 


248 


478.4 


293 


559.4 


159 


318.2 


204 


399-2 


249 


480.2 


: 294 


561.2 


160 


320.0 


205 


401.0 


250 


482.0 


i 295 


563.0 


161 


321.8 


206 


402.8 


251 


483.8 


> 296 


564.8 


162 


3236 


207 


404.6 


252 


485.6 


297 


566.6 


163 


325.4 


208 


406.4 


253 


487.4 


298 


568.4 


164 


327.2 


209 


408.2 


254 


489.2 


299 


570.2 


'^l 


329.0 


210 


410.0 


255 


491.0 


300 


572.0 


166 


330.8 


211 


41 1.8 


256 


492.8 


301 


573.8 


167 


332.6 


212 


413.6 


257 


494.6 


302 


575.6 


168 


334-4 


213 


415-4 


258 


496.4 


303 


577.4 


169 


336.2 


' 214 


417.2 


259 


498.2 


304 


579.2 


170 


338.0 


215 


419.0 


260 


500.0 


305 


581.0 


171 


339-8 


216 


420.8 


261 


501.8 


306 


582.8 


172 


341.6 


217 


422.6 


262 


503.6 


307 


584.6 


173 


343.4 


218 


424.4 


263 


505.4 


308 


586.4 


174 


345.2 


219 


426.2 


264 


507.2 


309 


588.2 


175 


347.0 


220 


428.0 


265 


509.0 


310 


590.0 


176 


348.8 


221 


429.8 


266 


510.8 


311 


591.8 


177 


350.6 


222 


431-6 


267 


512.6 


312 


593.6 


178 


352.4 


223 


433.4 


268 


514.4 


313 


595-4 


179 


354.2 ' 


224 


435.2 


269 


516.2 


314 


597.2 


180 


356.0 


225 


437.0 


270 


518.0 


315 


599.0 


181 


357.8 


226 


438.8 


271 


519.8 


316 


600.8 


182 


359.6 


227 


440.6 


272 


521.6 


317 


602.6 


183 


361.4 


228 


442.4 


273 


523.4 


318 


604.4 


184 


363.2 


229 


444.2 


274 


525.2 


319 


606.2 


185 


365.0 


230 


446.0 


275 


527.0 


320 


608.0 


186 


366.8 


231 


447.8 


276 


528.8 






187 


368.6 


232 


449.6 


277 


530.6 






188 


370.4 


233 


451.4 


278 


532.4 







AIR-THERMOMETERS. 



32s 



C 



t 




Air-Thermometers. 

Air-thermometers, or gas-thermometers, though inconvenient because 
bulky, are, by reason of the great expansiveness of air, superior to such as 
depend upon the expansion of liquids or solids, in point of delicacy and 
exactness. In any thermometer, whether liquid or gas, the indications 
depend jointiy upon the expansion by heat of the fluid substance, and that 
of the tube which holds it The expansion of mercury is scarcely seven 
times that of the glass tube within which it expands, and the exactness of 
its indications are interfered with by the variation in the expansiveness of 
glass of different qualities. In the gas-thermometer, on the contrary, the 
expansiveness of the gas is 160 times that of the glass, and the inequalities 
of the glass do not sensibly affect the indications of the instrument 

Gas-thermometers, or, as they are commonly called, air-thermometers, 
are designed either to maintain a constant pressure with a varying volume 
of air, or to maintain a constant volume of air while the pressure varies. 
In the first case. Fig. 119, the thermometer consists of a reservoir a, to be 
placed in the substance of which the temperature 
is to be ascertained; a tube d/, connected at a 
suitable distance by a small tube alf to the reservoir; 
a tube cd, open above, through which mercury is 
introduced into the instrument; a stop-cock r to 
open or close a communication — ist, between the 
tube dfzjoA the atmosphere; 2d, between the base 
of the tube cd and the atmosphere; 3d, between 
the two tubes df^ cd\ 4th, between both these 
tubes and the atmosphere. The tube df^ which is 
carefully gauged, answers the purpose of the gradu- 
ated tube of the mercury-thermometer, and receives 
the air driven over by expansion from the reservoir, 
at the same time that it is maintained at or near 
the temperature of the surrounding atmosphere. 
Thus the air is divided between the reservoir a and 
the tube df, of which the air in the former is at the ^* "9.-Air-Thcnnomcter. 

temperature of the substance under observation, and that in the latter is 
at the temperature of the atmosphere. These two portions of air support 
the same pressure, which qm at all times be approximated to that of the 
atmosphere by means of the cock r, through which the mercury is allowed 
to escape until it arrives at the same level in the two tubes. By means of 
a formula embracing the respective volumes of the two portions of air and 
the temperature of the atmosphere, the temperature of the substance under 
observation is determined. But it is apparent that, when applied as a 
pyrometer to the measurement of high temperatures — ^higher, that is to 
say, than the boiling point of mercury (676° F.) — ^the greater part of the air 
passes by expansion into the tube df^ leaving but a small remainder in the 
reservoir a. A serious objection to this is that the proportion of air which 
passes over into the tube df for a new increase of temperature is very 
small, and is with difficulty measured with sufficient precision. 

The second form of air thermometer, in which the pressure varies whilst 
the volume remains the same, was used by M. Regnault in his researches. 




326 



HEAT. 



f 



The temperature is measured by means of the increased elastic force of the 
inclosed air, and the instrument is both more convenient and more precise 
than that in which the volume varies, for at all temperatures the sensibility 
of the instrument is the same. At high temperatures the apparatus is liable 
to distortion under the pressure of the inclosed air; but this may be pre- 
vented, if needful, by introducing air of a lower than atmospheric- pressure at 
an ordinary temperature, even so low as one-fourth of an atmosphere; 
for, although the apparatus is less sensitive in proportion as the first supply 
of air is of less density and pressure, yet withal it is sufficiently sensitive. 
The thermometer, as employed by M. Regnault, is shown in Fig. 120. Two 

glass tubes, df^ cd, about half-an-inch bore, 
are united at the base by a stop-cock r. 
The tube cd is open above, and df is con- 
nected to the reservoir a by a small tube ab. 
The cover of the boiler in which the reser- 
voir is inclosed is shown at b, and the tubes 
are protected from the heat of the boiler by 
the partition c d. By means of a three-way 
connection, g^ and tube hy the connecting 
tube ab communicates with an air pump, 
by means of which the apparatus may be 
dried, and air or other gas supplied to it 
The first thing to be done is to completely 
dry the apparatus, and for this object, a little 
mercury is passed into the tube bd^ and the 
cock r is closed against it. The exhausting 
pump is then set to work to exhaust the 
tube, which is done several times, the air 
being slowly re-admitted after each exhaus- 
tion, after having been passed through a 
filter of pumice-stone in connection with 
the pump, saturated with concentrated sul- 
phuric acid to absorb moisture, and thus 
desiccate the air. During this part of the 
process, the reservoir is maintained at a 
temperature of 130° F., or 140° F., to insure 
complete desiccation. Next, the reservoir is plunged into melting ice, 
the two vertical tubes bd^ cd, are put into conynunication, and filled with 
mercury up to a suitable level /, marked on the tube bd. If it is desired 
to establish an internal pressure less than that of the atmosphere, the air 
is partially exhausted by means of the pump, the degree of exhaustion being 
recorded by the difference of level in the two tubes. The exhausting tube 
h is then hermetically sealed, and the mercury adjusted to the level /in the 
tube bd. 



G 



Fig. Z20. 



Pyrometers. 

Pyrometers are employed to measiu-e temperatures above the boiling 
point of mercury, about 676* F. They depend upon the change of form of 
either solid or gaseous bodies, liquids being necessarily inadmissible. 
Pyrometric estimations are of three classes : — First, those of which the 



PYROMETERS. 32/ 

indications are based upon the change of dimensions of a particular body, 
solid or gaseous — the pyrometer; second, those based on the heat imparted 
to water by a heated body; third, those which are based upon the melting 
points of metals and metallic alloys. 

Wedgwood^s pyrometer, invented in 1782, was founded on the property 
possessed by clay of contracting at high temperatures, an eflfect which is 
due partly to the dissipation of the water in clay, and subsequently to partial 
vitrification. The apparatus consists of a metallic groove, 24 inches long, 
the sides of which converge, being half-an-inch wide above and three-tenths 
below. The clay is made up into little cylinders or truncated cones, which 
fit the commencement of the groove after having been heated to low red- 
ness; their subsequent contraction by heat is determined by allowing them 
to slide from the top of the groove downwards till they arrive at a part of 
it through which they cannot pass. The zero point is fixed at the tempera- 
ture of low redness, 1077° F. The whole length of the. groove or scale is 
divided into 240 degrees, each of which was supposed by Wedgwood 
equivalent to 130° F., the other end of the scale being assumed to represent 
32,277° F. Wedgwood also assumed that the contraction of the clay was 
proportional to the degree of heat to which it might be exposed; but this 
assumption is not correct, for a long-continued moderate heat is found to 
cause the same amount of contraction as a more violent heat for a shorter 
period. Wedgwood's pyrometer is not employed by scientific men, because 
its indications cannot be relied upon for the reason just given, and also 
because the contraction of different clays under great. heat is not always the 
same. 

In Daniell's pyrometer the temperature is measured by the expansion of 
a. metal bar inclosed in a black-lead earthenware case, which is drilled out 
longitudinally to ^ inch in diameter and 7^ inches deep. A bar of 
platinum or soft iron, a little less in diameter, and an inch shorter than the 
bore, is placed in it and surmounted by a porcelain index i J^ inches long, 
kept in its place by a strap of platinum and an earthenware wedge. 
When the instrument is heated, the bar, by its greater rate of expansion 
compared with the black-lead, presses forward the index, which is kept in 
its new situation by the strap and wedge until the instrument cools, when 
the observation can be taken by means of a scale. 

The air-pyrometer. The principle and construction of the air-thermo- 
meter are directly applicable for pyrometric purposes, substituting a platinum 
globe for the glass reservoir already described, for resisting great heat, and 
as large as possible. The' chief cause of uncertainty is the expansion of 
the metal at high temperatures. 

The second means of estimation is best represented by the " pyrometer " 
of Mr. Wilson, of St. Helen's. He heats a given weight of platinum in the 
fire of which the temperature is to be measured, and plunges it into a 
vessel containing twice the weight of water of a known temperature. 
Observing the rise of temperature in the water, he calculates the tempera- 
ture to which the platinum was subjected, in terms of the rise of tempera- 
ture of the water, the relative weights of the platinum and the water, and 
their specific heats. In fact, the elevation of the temperature of the 
water is to that of the platinum above the original temperature of the water 
in the compound ratio of the weights and specific heats inversely; that is 
to say, that the weights of the platinum and the water being as i to 2, and 



328 HEAT. 

their specific heats as .0314 to i, the rise of temperature of the water is to 
that of the platinum as i x .0314 to 2 x i, or as i to 63.7, and the rule for 
finding the temperature of the fire is to multiply the rise of temperature of 
the water by 63. 7, and add its original temperature to the product. The 
sum is the temperature of the fire, subject to correction for the heat 
absorbed by the thermometer in the water, and by the iron vessel contain- 
ing the water, and the heat retained by the platinum. The correction is 
estimated by Mr. Wilson at iV^j taking the weight of water at 2000 grains, 
and that of the platinum 1000 grains, and it may be allowed for by increas- 
ing the above-named multiplier by -rrth, to 67.45. 

Mr. Wilson proposed that for general practical purposes a small piece of 
Stourbridge clay be substituted for platinum, to lessen the cost of the 
apparatus. With a piece of such clay, weighing 200 grains, and 2000 
grains of water, he found that the correct multiplier was 46. 

The third means of estimation, based on the melting points of metals 
and metallic alloys, is applied simply by suspending in the heated medium 
a piece of metal or alloy of which the melting point is known, and, if 
necessary, two or more pieces of different melting points, so as to ascertain, 
according to the pieces which are melted and those which continue in the 
solid state, within certain limits of temperature, the heat of the furnace. A 
list of melting points of metals and metallic alloys is given in a subsequent 
chapter. 

Luminosity at High Temperatures. 

The luminosity or shades of temperature have been observed by M. 
Pouillet by means of an air-pyrometer to be as follows : — 

Shaob. Tempsraturs, Temperature, 

Centigrade. Fahrenheit. 

Nascent Red 525° 977° 

Dark Red 700 1292 

Nascent Cherry Red 800 1472 

Cherry Red 900 1652 

Bright Cherry Red 1000 1832 

Very Deep Orange iioo 2012 

Bright Orange. 1200 2192 

White 1300 2372 

"Sweating" White 1400 2552 

Dazzling White 1500 2732 

A bright bar of iron, slowly heated in contact with air, assumes the 
following tints at annexed temperatures (Claudel) : — 

Centigrade. Fahrenheit 

1. Cold iron at about 12*' or 54° 

2. Yellow at 225 437 

3. Orangeat 243 473 

4. Red at 265 509 

5. Violetat 277 531 

6. Indigo at 288 550 

7- Blue at 293 559 

8. Green at 332 630 

9. Oxide Gray (gris cToxyde) at 400 752 



MOVEMENTS OF HEAT. 329 

MOVEMENTS OF HEAT. 

When two bodies in the neighbourhood of each other have unequal 
temperatures, there exists between them a transfer of heat from the hotter 
of the two to the other. The tendency to an equalization, or towards an 
equilibrium, of temperatures in this way is universal, and the passage of 
heat takes place in three ways : by radiation, by conduction, and by con- 
vection or carriage from one place to another by heated currents. 

Radiation of Heat from Combustibles. 

It is a common assumption that the radiation of heat from combustibles 
is relatively very small in comparison with the total quantity of heat 
evolved. Holding the hand near the flame of a candle, laterally, the 
radiant heat, which is the only heat thus experienced, is much less than the 
heat experienced by the hand when held above the flame, which is the heat 
by convection of the hot current of air which rises from the flame. But it 
is to be noted that, whilst the radiant heat is dissipated all round the flame, 
the diameter of the upward current is little more than that of the flame, 
and the conveyed heat is therefore concentrated in a small compass. 

M. Peclet, by means of a simple apparatus, consisting of a cage suspend- 
ing the combustible within a hollow cylinder filled with water in an annular 
space, ascertained that the proportion of the total 'heat radiated from 
clifrerent combustibles was as follows : — 

Radiant heat from wood nearly }(. 

Do. do. wood charcoal „ J?. 

Do. do. oil „ '/j. 

These values serve to show that radiation of heat is considerable, and that 
flameless carbon radiates much more than flame, though the proportion of 
heat radiated from fuels depends very much upon the disposition of the 
material and the extent of radiating surface. 

With respect to heated bodies, apart from combustibles as such, the 
radiation or emission of heat implies the reverse process of absorption, and 
the best radiators are likewise the best absorbents of heat. All bodies 
possess the property of radiating heat The heat rkys proceed in straight 
lines, and the intensity of the heat radiated from any one source of heat 
becomes less as the distance from the source of heat increases, in the 
inverse ratio of the square of the distance. That is to say, for example, 
that at any given distance from the source of radiation, the intensity of the 
radiant heat is four times as great as it is at twice the distance, and nine 
times as great as it is at three times the distance. 

The quantity of heat emitted by radiatioh increases in some proportion 
with the difference of temperatures of the radiating body and the surrounding 
medium, but more rapidly than the simple proportion for the greater differ- 
ences; and the quantity of heat, greater or less, emitted by bodies by radiation 
under the same circumstances is the measure of their radiating p<nver. 

Radiant heat traverses air without heating it. 

When a polished body is struck by a ray of heat, it absorbs a part of the 
heat and reflects the rest The greater or less proportion of heat absorbed 
by the body is the measure of its absorbing power, and the reflected heat is 
the measure of its reeding power. 



330 



HEAT. 



When the temperature of a body remains constant it indicates that the 
quantity of heat emitted is equal to the quantity of heat absorbed by the 
body. The reflecting power of a body is the complement of its absorbing 
power; that is to say, that the sum of the absorbing and reflecting powers of 
all bodies is the same, which amounts to this, that a ray of heat striking a 
body is disposed of by absorption and reflection together, that which is not 
absorbed being necessarily reflected. 

For example, the radiating power of a body being represented by 90, the 
reflecting power is also 90, and the absorbing power is lo, supposing that 

Table No. 106. — Comparative Radiating or Absorbent and 

Reflecting Powers of Substances. 



Substance. 



Lamp Black 

Water 

Carbonate of Lead 

Writing Paper 

Ivory, Jet, Marble 

Isinglass 

Ordinary Glass 

China Ink 

Ice 

Gum Lac 

Silver Leaf on Glass 

Cast Iron, brightly polished 

Mercury, about 

Wrought Iron, polished 

Zinc, polished 

Steel, polished 

Platinum, a little polished 

Do. deposited on Copper . . . 

Do. in Sheet 

Tin 

Brass, cast, dead polished 

Do. hammered, dead polished., 

Do. cast, bright polishefl 

Do. hammered, bright polished 

Copper, varnished 

Do. deposited on iron 

Do. hammered or cast 

Gold, plated 

Do. deposited on polished Steel 
Silver, hammered, polished bright 

Do. cast, polished bright 



Powers. 



Radiating or 
Absorbing. 


Reflecting. 


100 





100 





100 





98 


2 


93 to 98 


7 to 2 


91 


9 


90 


10 


85 


15 


85 


15 


72 


28 


27 


73 


25 


75 


23 


77 


23 


77 


19 


Si 


17 ' 


83 


24 


76 


17 


83 


17 


83 


15 


85 


II 


89 


9 


91 


7 


93 


7 


93 


14 


86 


7 


93 


7 


93 


5 


95 


3 


97 


3 


97 


3 


97 



MOVEMENTS OF HEAT. 



331 



the total •quantity of heat which strikes the body is represented by 100. 
The reflecting power of soot is sensibly «/7, and its absorbing and radiating 
powers are 100. 

The absorbing power varies with the nature of the source of heat, with 
the condition of the substance, and with the inclination of the direction of 
the heat radiated upon the body. That of a metallic surface is so much 
the less, and consequently the reflecting power is so much the more, in 
proportion as the surface is better polished. 

The reflecting power of metals, according to MM. de la Provostaye and 
Desains, is practically the same, when the angle of incidence, that is the 
angle at which the rays of heat strike the surface, is less than 70® of inclina- 
tion with the surface; but for greater angles, approaching more nearly to 
90°, perpendicular to the surface, it sensibly diminishes. 

For example, at angles of from 75 to 80 degrees, the reflecting power is 
only 94 per cent, of what it is under the smaller angles of incidence. 

The table No. 106 contains the radiating and absorbing powers and the 
reflecting powers of various substances. (Leslie^ De la Provostaye and 
JDesainSy and Melloni,) 

The reflecting power of glass has been found to be the same for heat and 
for light 

Conduction of Heat. — Conduction is the movement of heat through sub- 
stances, or from one substance to another in contact with it. The table 
No. 107 contains the relative internal conducting power of metals and earths, 
according to M. Despretz. A body which conducts heat well is called a 
good conductor of heat; if it conducts heat slowly, it is a bad conductor of 
heat. Bodies which are finely fibrous, as cotton, wool, eider-down, wadding, 
finely divided charcoal, are the worst conductors of heat. Liquids and 
gases are bad conductors; but if suitable provision be made for the free 
circulation of fluids they may abstract heat very quickly by contact with 
heated surfaces, acting by convection. 

Convection of Heat, — Convected or carried heat is that which is trans- 
ferred from one place to another by a current of liquid or gas : for example, 
by the products of combustion in a furnace towards the heating surface 
in the flues of a boiler. 



Table No. 107. — Relative Internal Conducting Power of Bodies. 



Substance. 


Relative conducting 
power. 


t 

Substance. 


Relative conducting 
power. 


Gold 


1000 
981 

973 
892 

749 
562 

374 


Zinc 


363 

304 
180 

24 
12 

II 


Platinum ; 


Tin 


Silver 


Lead 


Copper 


Marble 


Brass 


Porcelain 


Cast Iron 


Terra Cotta 


Wrought Iron 







THE MECHANICAL THEORY OF HEAT. ' 

Heat and mechanical force are identical and convertible. Independently 

of the medium through which heat may be developed into mechanical 
action, the same quantity of heat is resolved into the same total quantity 
of work. The English unit of heat is that which is required to raise the 
temperature of i lb. of water r degree Fahr. If 2 lbs. of water be raised 
1 degree, or i lb. be raised 2 degrees in temperature, the expenditure of 
f heaCj and to express the 
lies between the unit of 
le foot-pound, on the other 
; of the numerical relation 
s obtained by the following 
iptator. Fig. izi, consisting 
of a vertical shaft carry- 
ing a brass paddle-wheel, 
of which the paddles re- 
volved between station- 
ary vanes, which served 
to prevent the liquid in 
the vessel from being 
bodily whirled in the 
direction of rotation. 
The vessel was filled 
with water, and the agi- 
tator was made to revolve 
bymeans of a cord wound 
round the upper part of 
the shaft, and attached 
to a weight which de- 
scended m front of a 
scale, by which the work 
done was measured. 
found that the heat com- 
ited to one pound-degree 
expended in producing it. 
at was capable of raising 
jnical equivalent of heat, 
;n as 772 foot-pounds for 
nedium, and it yielded die 

alent for different thermo- 

bot-pounds, 

;s(say424)tilogrammetres. 
.60 (say 1390) foot-pounds, 
[lish thermal units — about 



MECHANICAL THEORY OF HEAT. 333 

According to the mechanical theory of heat, in its general form, heat, 
mechanical force, electricity, chemical affinity, light, and sound, are but 
different manifestations of motion. Dulong and Gay Lussac proved by 
their experiments on sound, that the greater the specific heat of a gas, the 
more rapid are its atomic vibrations. Elevation of temperature does not 
alter the rapidity but increases the length of their vibrations, and in con- 
sequence produces "expansion " of the body. All gases and vapours are 
assumed to consist of numerous small atoms, moving or vibrating in all 
directions with great rapidity; but the average velocity of these vibrations 
can be estimated when the pressure and weight of any given volume of the 
gas is known, pressure being, as explained by Joule, the impact of those 
numerous small atoms striking in all directions, and against the sides of the 
vessel containing the gas. The greater the number of these atoms, or the 
greater their aggregate weight, in a given space, and the higher the velocity, 
the greater is the pressure. A double weight of a perfect gas, when con- 
fined in the same space, and vibrating with the same velocity — that is, 
having the same temperature — gives a double pressure; but the same weight 
of gas, confined in the same space, will, when the atoms vibrate with a 
double velocity, give a quadruple pressure. An increase or decrease of 
temperature is simply an increase or decrease of molecular motion. When 
the piston in the cylinder yields to the pressure of steam, the atoms will 
not rebound from it with the same velocity with which they strike, but will 
return after each succeeding blow, with a velocity continually decreasing 
as the piston continues to recede, and the length of the vibrations will be 
diminished. The motion gained by the piston will be precisely equivalent 
to the energy, heat, or molecular motion lost by the atoms of the gas; 
and it would be as reasonable to expect one billiard ball to strike and give 
motion to another without losing any of its own motion, as to suppose that 
the piston of a steam-engine can be set in motion without a corresponding 
quantity of energy being lost by some other body. 

In expanding air spontaneously to a double volume, delivering it, say, 
into a vacuous space, it has been proved repeatedly that the air does not 
appreciably fall in temperature, no external work being performed; but that, 
on the contrary^ if the air at a temperature, say, of 230° F., be expanded 
against an opposing pressure or resistance, as against the piston of a cylinder, 
giving motion to it and raising a weight or otherwise doing work, the tem- 
perature will fall nearly 170® F. when the volume is doubled, that is from 
230° F. to about 60® F., and, taking the initial pressure at 40 lbs., 
the final pressure would be 15 lbs. per square inch. 

When a pound weight of air, in expanding, at any temperature or pressure, 
raises 130 lbs. one foot high,' it loses i® F. in temperature; in other 
words, this pound of air would lose as much molecular energy as would 
equal the energy acquired by a weight of one pound falling through a 
height of 130 feet. It must, however, be remarked that but a small portion 
of this work — 130 foot-pounds — can be had as available work, as the heat 
which disappears does not depend on the amount of work or duty realized, 
but upon the total of the opposing forces, including all resistance from any 
external source whatever. When air is compressed the atmosphere descends 
and follows the piston, assisting in the operation with its whole weight; and 
when air is expanded the motion of the piston is, on the contrary, opposed 
by the whole weight of the atmosphere, which is again raised. Although, 



334 HEAT. 

therefore, in expanding air, the heat which disappears is in proportion to 
the total opposing force, it is much in excess of what can be rendered 
available; and, commonly, where air is compressed the heat generated is 
much greater than that which is due to the work which is required to be 
expended in compressing it, the atmosphere assisting in the operation. 

Let a pound of water, at a temperature of 212** F., be injected into a 
vacuous space or vessel, having 26.36 cubic feet of capacity — the volume 
of one pound of saturated steam at that temperature — ^and let it be evapor- 
ated into such steam, then 893.8 units of heat would be expended in the 
process. But if a second pound of water, at 212°, be injected and evapor- 
ated at the same temperature, under a uniform pressure of 14.7 lbs. per 
square inch, being the pressure due to the temperature, the second pound 
must dislodge the first, supposing the vessel to be expansible, by repelling 
'that pressure; and this involves an amount of labour equal to 55,800 foot- 
pounds (that is, 14.7 lbs. x 144 square inches x 26.36 cubic feet), and an 
additional expenditure of 72.3 units of heat (that is, 55,800 -f- 772), making 
a total, for the second pound, of 965. i units. 

Similarly, when 1408 units of heat are expended in raising the tempera- 
ture of air under a constant pressure, 1000 of these units increase the 
velocity of the molecules, or produce a sensible increase of temperature ; 
while the remaining 408 units, which disappear as the air expands, are 
directly consumed in repelling the external pressure for the expansion of 
volume. 

Again, if steam be permitted to flow from a boiler into a comparatively 
vacuous space without giving motion to another body, the temperature of 
the steam entering this space would rise higher than that of the steam in 
the boiler. Or, suppose two vessels, side by side, one of them vacuous and 
the other filled with air at, say, two atmospheres; if a communication be 
opened between them, the pressure becomes the same in both. But the 
temperature would fall in one vessel and rise in the other; and although 
the air is expanded in this manner to double its first volume, there would 
not, on the whole, be any appreciable loss of heat, for if the separate por- 
tions of air be mixed together, the resulting average temperature of the 
whole would be very nearly the same as at first It has been proved 
experimentally, corroborative of this statement, that the quantity of heat 
required to raise the temperature of a given weight of air, to a given extent, 
is the same, irrespective of the density or the volume of the air. Regnault 
and Joule found that to raise the temperature of a pound of air, whether 
I cubic foot or 10 cubic feet in volume, the same quantity of heat was 
expended. 

In rising against the force of gravity steam becomes colder, and it par- 
tially condenses while ascending, in the effort of overcoming the resistance 
of gravity. For instance, a column of steam weighing, on a square inch of 
base, 250.3 lbs., that is to say, having a pressure of 250.3 lbs. per square 
inch, would, at a height of 275,000 feet, be reduced to a pressure of i lb. 
per square inch, and, in ascending to this height, the temperature would 
fall from 401° to 102° F., while, at the same time, nearly 25 per cent 
of the whole vapour would be precipitated in the form of water, unless it 
were supplied with additional heat while ascending. 

If a body of compressed air be allowed to rush freely into the atmosphere, 
the temperature falls in the rapid part of the current, by the conversion of 



EXPANSION BY HEAT. 335 

heat into motion, but the heat is almost all reproduced when the motion 
has quite subsided. From recent experiments, it appears that nearly 
similar results are obtained from the emission of steam under pressure. 

When water falls through a gaseous atmosphere, its motion is constantly 
retarded as it is brought into collision with the particles of that atmosphere, 
and by this collision it is partly heated and partly converted into vapour. 

If a body of water descends freely through a height of 772 feet, it acquires 
from gravity a velocity of 223 feet per second; and, if suddenly brought to 
rest when moving with this velocity, it would be violently agitated, and 
would be raised one degree of temperature. But suppose a water-wheel, 
772 feet in diameter, into the buckets of which the water is quietly dropped; 
when the water descends to the foot of the fall, and is delivered gently into 
the tail-race, it is not sensibly heated. The greatest amount of work it is 
possible to obtain from water falling from a given level to a lower level is 
expressible by the weight of water multiplied by the height of the fall. 

These illustrative exhibitions of the nature and reciprocal action of heat 
and motive power, show that the nature and extent of the change of tem- 
perature of a gas while expanding depend nearly altogether upon the cir- 
cumstances under which the change of volume takes place. 



EXPANSION BY HEAT. 

All bodies are expanded by the application of heat, but in different 
degrees. Expansion is measurable in three directions : — Length, breadth, 
and thickness; and it may be measured as linear expansion, in one direc- 
tion; as superficial expansion, in two directions; or as cubical expansion, 
in three directions. Linear expansion, or the expansion of length, is that 
which will be exposed in the following tables for solids and liquids. The 
expansion of gases is measured cubically, by volume. 

Superficial expansion, it may be added, is twice the linear expansion, and 
cubical expansion is three times the linear expansion. That is to say, the 

additional volume by expansion in two direc- 
, tions, as in length and breadth, is twice the 
I additional volume in one direction; and the 
: additional volume in three directions is three 
I times that in one direction. For example, 
j take a solid cube abcdefg\ the expansion in 
' one direction ea^ on the face abed, is, say, 
equal to that indicated by the dot lines pro- 
jected from that face, and the volume by 
expansion is equal to the extension of the 
surface abed thus projected. In each of the 
two other directions, da, upwards, and ab, 
*• "^ laterally, the volume by. expansion is the same 

as that of the expansion on the face abed. Consequently, the total 
increase of volume by expansion, as measured cubically, in the three 
directions of length, breadth, and thickness, is three times the increase of 
volume in one direction singly; and, as measured superficially, in two of 
these directions, it is twice the increase of volume in one direction. 



336 



HEAT. 



Table No. io8. — Linear Expansion of Solids by Heat, between 

32° AND 212*" F. 



METALS. 



} 



Zinc, sheet 

Do., forged 

Lead 

Zinc 8+1 tin, slightly ham 

mered, 

White Solder: — tin i -♦- 2 lead. 

Tin, grain 

Tin 

Silver 

Speculum metal 

Brass 

Copper 

Gun Metal: — 16 copper + i tin 

8 copper + i tin 

Yellow Brass : — Rod 

Do. Trough form.. 

Gold:— 

Paris standard,' annealed 
Do. unannealed 

Bismuth 

Iron, forged 

Do. wire 

Steel, rod, 5 feet long 

Do. tempered 

Do. not tempered 

Cast Iron, rod, 5 feet long 

Antimony 

Palladium 

Platinum 



Expansion 
between m' 
and 213* F. 
in common 

fractions. 



1/ 

■/3s. 

■/: 

■/ 
■/ 



37a 



399 



V 



403 

524 

753a 
V58X 

V5a4 

Vsaa 

Vsa8 



V. 



/66x 
64s 
V719 
V8x9 
Vsia 

V874 

y8o7 

'/901 
^/9a3 
/xooo 

7x167 



Expansion 

between 32* 

and 2x2' F. 

in a length 

=ioa 



length = xoo. 

29416 

31083 

28484 

26917 

25053 

24833 
21730 

19075 

19333 
18782 

17220 

19083 

18167 

18930 

18945 

I5153 
I5516 

I3917 
12204 

12350 

1 1450 

12396 

10792 

moo 
10833 
1 0000 
08570 



Expansion 

between 32' 

and 2X2* F. 

in a length' 

of xo feet. 



inch. 

•353 
.374 
.342 

.322 

.301 
.298 
.260 
.229 
.232 
.225 
.207 
.229 
.218 
.227 
.227 

.181 
.186 
.167 
.146 
.148 

•137 
.149 

.130 

•133 
.130 

.120 

.103 



Expansion 

for I F. in a 

length of TOO 

fecL 



inch. 
.0196 
.0207 
.0190 

.0179 

.0167 
.0166 

.0145 
.0127 

.0130 

.0125 

.0115 

.0127 

.0121 

.0126 

.0126 

.0101 

.0103 

.00928 

.00814 

.00823 

.00763 

.00826 

.00719 

.00740 

.00722 

.00667 

.00571 



From 0° to 300° C. 
(32° F. to 572° F.) 



Copper... 

Iron 

Platinum. 



/ 0° to 100° C. 
( 0° to 300° C. 
/ 0° to loo** C. 
\ 0° to 300° C. 

{ 



0° to 100° C. 
o** to 300'' C. 



1/582 

V846 
V68x 
71x03 
Vxo89 



.17182 
.18832 
.11821 
.14684 
.08842 
.09183 



.206 
.226 
.142 
.176 
.106 
.III 



.0115 

.00418 

.00788 

.00326 

.00589 

.00204 



LINEAR EXPANSION OF SOLIDS BY HEAT. 



337 



Table No. io8 {continued). 



GLASS. 



Flint Glass 

French Glass, with lead 

Glass tube, without lead 

Glass of St. Gobain 

Barometer tubes (Smeaton). 

Glass tube (Roy) 

Glass rod, solid (Roy) 

Glass (Dulong and Petit) . . . 

Do. (o° to 200° C.) 

Do. (o'' ta3oo°C.) 

Ice 



Expansion 

between 32* 

and 213* F. 

in common 

fractions. 



/1248 

Am7 
/1090 
/zx2a 

/"75 
/1289 

/"37 
V1032 

V987 



Expansion 

between 32* 

and 2X2° r. 

in a length 

=ioa 



.08117 
.08720 

.09175 
.08909 

.08333 

.07755 
.08083 

.08613 

.09484 

.10108 



Expansion 
between 32* 
and 2X2* F. 
in a length 

of xo feet. 



inch. 
.0974 
.105 
.110 
.107 
.100 
.0931 
.0970 
.103 
.114 
.121 



sion 



Expansi 
for X F. in a 
length of xco 
feet. 



inch. 
.00541 
.00581 
.00612 
.00594 

.00555 
.00517 

.00539 

.00574 

.00632 

.00674 



■0333 



STONES. 



Granite 

Do 

Cky-^late 

Do. 

York paving. 

Micaceous sandstone. 

Do. do. 

Do. do. 

Do. do. 

Do. do. 

Do. do. 

Carrara marble 

Sost do 

Stock Brick , 



Initial , 
Temperature. 



Final 
Temperature. 



45^ 

45 
46 

46 

46 

52 
52 
52 
52 
45 
45 
32 
32 
52 



F. 



220 
100 

87 

104 

95 
200 

200 

150 
100 

100 

260 

212 

212 

260 



F. 



Expansion 

in a length 

=xoo. 



length = too. 
.2916 
.0416 
.0416 
.0693 
.1695 

.1736 
.1041 
.0832 
.0520 
.0416 
.1458 
.0849 
.0568 
.2500 



Expansion 

for X F. in a 

length of xoo 

feet. 



inch. 
.0200 
.00908 
.0122 

.0143 

.0415 
.0141 

.00844 

.0102 

.01300 

.00908 

.00814 

.00566 

.00380 

.00144 



25" 



338 HEAT. 

Speaking exactly, the cubical expansion is rather less than three times, 
and the superficial expansion tather less than twice, the linear expansion ; 
for, in fact, the expanded comers of the body are carried out to the full 
square figure, and have not the entering angles shown in the figure, and 
there is, in this way, a certain overlapping of the strata of expansion at the 
ends, sides, and top. 

The same kind of demonstration applies to bodies of any other than a 
cubical shape. 

A hollow body expands by heat to the same extent as if it were a solid 
body having the same exterior dimensions. 

The rate of expansion of solids from the freezing point to the boiling 
point of water, 32° to 212*' R, is sensibly uniform. 

The table. No. 108, gives the linear expansion of a number of metals, 
and of glass, between the freezing and boiling points;, and of ice for one 
degree, and of stones for various intervals of temperature. Authorities: — 
Laplace and Lavoisier, Smeaton, Roy, Troughton, Wollaston, Dulong and 
Petit, Froment, Rennie. 

Zinc is the most expansible of the metals; it expands fully one-third 
per cent, or as much as Vs^iSt part of its length, when heated from 32° F. 
to 212° F. Iron expands about one-seventh to one-eighth per cent.; and 
cast-iron and platinum about one-tenth per cent. The expansion of metals 
proceeds at a less rate above the boiling point than below it. Ice expands 
at the rate of V36,oooth of its length for one degree Fahrenheit; which, for 
180°, would be Vaooth of its length, — greatly more than that of any metal. 

Expansion of Liquids. 

The measurement of the expansion of liquids by the application of heat 
cannot well be taken lineally; that is, as linear expansion, in th6 sense 
in which the expansion of solids is observed. For liquids must be con- 
tained in vessels, which only admit of expansion in one direction, seeing 
that the liquid is limited by the bottom and sides of the vessel, which 
throw the whole of the expansion or enlargement of volume upwards. The 
observations on the expansion of liquids, therefore, though measured in 
one direction only, necessarily indicate the cubical expansion or total 
enlargement of volume. But, of course, it is easy to reduce the expansion 
of a liquid for comparison with the linear expansion of a solid by taking 
one-third of the observed measurement. 

When the temperature of water at the freezing point, 32° F., is raised, 
the water does not at first expand, but, on the contrary, contracts in volume 
until the temperature is raised to 39°.! F., which is 7.1 degrees above the 
freezing point. This is called "the temperature of maximum density." 
From this point water expands as the temperature rises, until, at 46° F., it 
regains its initial volume, that is, the volume at 32° F. Thence, it con- 
tinues to expand until it reaches the boiling point, 212° F., under one 
atmosphere. Passing this point upwards, if the pressure be suitably 
increased, water continues to expand with a rise of temperature. 

The cubical expansion of water when heated from 32® to 212° F. is 

.0466; that is, the volume is increased from i at 32° F. to 1.0466 at 212** 

F. This expansion is rather more than 4^ per cent, or between '/aiSt and 

'/aad part of the volume at 32°. The expansion of water increases in a 



EXPANSION OF LIQUIDS BY HEAT. 



339 



Table No. 109. — Expan^on and Density of Pure Water, 

FROM 32** TO 390° F. 

(Calculated by means of Rankine's approximate formiila. ) 



Tempera- 
ture. 


Comparative 
Volume. 


Comparative 
Density. 


Density, 
or weight of 
X cubic foot. 


Weight of 
X gallon. 


Remarkable Temperatures. 


Fahr. 


Water at 32' 
= 1. 


Water at 32' 
= I. 


Pounds. 


Pounds. 


/ 


32^ 


1. 00000 


1. 00000 


62.418 


lO.OIOI 


Freezing point. 


35 


0.99993 


1.00007 


62.422 


10.0103 




39-1 


0.99989 


1. 000 1 1 


62.425 


IO.OII2 


Point of maximum density. 


40 


0.99989 


1. 0001 1 


62.425 


IO.OII2 




45 


0.99993 


1.00007 


62.422 


10.0103 


- 


46 


1. 00000 


1. 00000 


62.418 


lO.OIOI 


f Same volume and density 
1 as at the freezing point 


50 


1. 00015 


0.99985 


62.409 


10.0087 


(Weight taken for ordi- 
\ nary calculations. 


52.3 


1.00029 


0.99971 


62.400 


10.0072 


55 


1.00038 


0.99961 


62.394 


10.0063 




60 


1.00074 


0.99926 


62.372 


10.0053 


1 


62 


I.OOIOI 


0.99899 


62.355 


10.0000 


Mean temperature. 


65 


I.00II9 


0.99881 


62.344 


9.9982 




70 


I.OOI60 


0.99832 


62.313 


9-9933 




75 


1.00239 


0.99771 


62.275 


9.9871 




80 


1.00299 


0.99702 


62.232 


9.980 




85 


1.00379 


0.99622 


62.182 


i 9-972 




90 


1.00459 


0-99543 


62.133 


9.964 




95 


1.00554 


0.99449 


62.074 


9-955 




100 


1.00639 


0-99365 


62.022 


9-947 


1 Temperature of conden- 
( ser water. 


105 


1.00739 


0.99260 


61.960 


9-937 




no 


1.00889 


0.991 19 


61.868 


9.922 


• 


"5 


1.00989 


0.99021 


61.807 


9-913 


• 


120 


I.01I39 


0.98874 


61.715 


9.897 




125 


1. 01239 


0.98808 


61.654 


9.887 




130 


1. 01390 


0.98630 


61.563 


9-873 




135 


I-OI539 


0.98484 


61.472 


9-859 




140 


1.01690 


0-98339 


61.381 


9.844 




145 


1. 01839 


0.98194 


61.291 


9.829 




150 


1. 01989 


0.98050 


61.201 


9.815 




155 


1. 02164 


0.97882 


61.096 


9-799 




160 


1.02340 


0.97714 


60.991 


9.781 




165 


1.02589 


0.97477 


60.843 


9-757 




170 


1.02690 


0.97380 


60.783 


9-748 




17s 


1.02906 


0.97193 


60.665 


9-728 




180 


1.03 100 


0,97006 


60.548 


9.711 




185 


1.03300 


0.96828 60.430 


9.691 





340 



HEAT. 



Table l^o, io<) {continued). 
(Calculated by means of Rankine's approximate formula.) 



Tempera- 
ture. 


Comparative 

Volume. 


Comparative 
Density. 


Density, 
or weight of 
I cubic foot. 


Weight of 
z gallon. 


Remarkable Temperatures. 


Fahr. 


Water at 32* 
= I. 


Water at 33* 
= I. 


Pounds. 


Pounds. 




190 


1.03500 


0.96632 


60.314 


9.672 




195 


1.03700 


0.96440 


60.198 


9.654 




200 


1.03889 


0.96256 


60.081 


9.635 




205 


1. 0414 


0.9602 


59.93 


9.61I 




210 


1.0434 


0.9584 


59.82 


9.594 




212 


1.0444 


0.9575 


59.76 


9-584 


Boiling point; by formula j 


212 


1.0466 


0.9555 


59.64 


9.565 


r Boiling point; by direct 
( measurement 


230 


1.0529 


0.9499 


59-36 


9.520 




250 


1.0628 


0.941 1 


58.75 


9.422 




270 


1.0727 


0.9323 


58.18 


9.331 




290 


1.0838 


0.9227 


57.59 


9.236 


( Temperature of steam of 
I 50 lbs. effective pres- 
( sure per square inch. 


298 


1.0899 


0.9175 


57.27 


9.185 






















I Temperature of steam of 


338 


I.III8 


0.8994 


56.14 

• 


9.004 


< 100 lbs. effective pres- 
( sure per square inch. 
i Temperature of steam of 


366 


I.I3OI 


0.8850 


55.29 


8.867 


< 150 lbs. effective pres- 
( sure per square inch. 
i Temperature of steam of 
-| 205 lbs. effective pres- 
( sure per square inch. 


390 


1. 1444 


0.8738 


54.54 


8.747 













greater ratio than the temperature. The annexed table No. 109 shows 
approximately the cubical expansion, comparative density, and comparative 
volume of water for temperatures between 32° and 212° F., calculated by 
means of an approximate formula constructed by Professor Rankine as 
follows : — 



D, nearly = ■? — ° — 

^ / + 461 5 



00 



(I) 



500 / + 461 



in which 00=62.425 lbs. per cubic foot, the maximum density of water, 
and D, = its density at a given temperature / F. 

Rule. — To find approximately the density of water at a given temperaturCy 
the maximum density being 62.425 lbs, per cubic foot. To the given tempera- 
ture in Fahrenheit degrees, add 461, and divide the sum by 500. Again, 
divide 500 by that sum. Add together the two quotients, and divide 
124.85 by the sum. The final quotient is the density nearly. 



EXPANSION OF LIQUIDS BY HEAT. 34 1 

The results given by this rule are very nearly exact for the lower tempera- 
tures, but for the higher temperatures they are too great. For 212° F. the 
density of water by the rule is 59.76 lbs. per cubic foot, but it is actually 
only 59.64 lbs., showing an error of about Vsooth part in excess. 

From the table it appears that the density of water at 46° F., or about 
8° C, is the same as at the freezing point, 32*^ F., and that the temperature 
of maximum density, 39°. i F., or 4° C., lies midway between those tempera- 
tures. The expansion of water towards and down to the freezing point is 
Vgoooth part of the volume at the temperature of maximum density. It 
would appear that in thus expanding from 39^.1 F. downwards, the particles 
of water enter on a preparatory stage of separation, anticipating the still 
further separation which ensues on the conversion of water into the solid 
state; for ice is considerably lighter than water and floats on it, and its 
density is little more than nine-tenths that of water. 

In passing upwards from the freezing point towards higher temperatures, 
the increase of volume of watef by expansion, in parts of the volume at 
the freezing point, is as follows : — 

Expansion in 

parts of the volume 

at 32' F. 

at 52°.3 F. corresponding to the weight per cubic foot 

(62.4 lbs.) usually taken for ordinary calcu- per cent 

latio ns .03 

at 62° the mean temperature .10 

at 1 00° the temperature of condenser water. .64 

at 212° the boiling point 4.66 

at 298° the temperature of steam of 50 lbs. effective 

pressure per square inch 9.0 

at 338** the temperature of steam of 100 lbs. effective 

pressure per square inch 1 1.2 

at 366° the temperature of steam of 150 lbs. effective 

pressure per square inch 13.0 

at 390° the temperature of steam of 205 lbs. effective 

pressure per square inch 14.4 

The expanded volume of some liquids from 32*^ to 212° F. is given in 
table No. no; that is, the apparent expansion as seen through glass. It 
is shown that alcohol and nitric acid are the most expansible, and water 
and mercury the least; the former expand one-ninth of their initial volume, 
and of the latter, water, as already stated, expands ^/aad part, and mercury 
Yesth part of their initial volumes respectively. Observations on the 
absolute expansion of mercury are added, and they show that whereas the 
apparent expansion in glass is 'As^h part, the real expansion is ^/^s^h part 
of the initial volume. 

No other liquid besides water has a point of maximum density ; that is, 
a point higher than the freezing point of the liquid. 



342 



HEAT. 



Table No. no. — Expansion of Liquids by Heat, from 3 2*^ to 212° F. 

Apparent Expansion, in Glass. 



Liquid. 



Alcohol 

Nitric Acid 

Olive Oil 

Linseed Oil 

Turpentine 

Sulphuric Ether 

Hydrochloric Acid (density 1.137) 

Sulphuric Acid (density 1.850) 

Water saturated with Sea Salt 

Water 

Mercury 



Volume at aia" F. 



volume at 3a* F.=i. 
I.IIOO 
I.IIOO 
1.0800 
1.0800 
1.0700 
1.0700 
1.0600 
1.0600 
1.0500 
1.0466 
I.OI54 



Ex]>ansion in 
Vulga^ Fractions. 



■/, 



18 



volume at 33* F.=:i. 

} 

V. 

Vm 
Vm 

V.7 

■■',: 

•A 
'As 



ao 



Absolute Expansion of Mercury. 



Volume at 
aia* F. 



Mercury, from 32* to 212** F. ( g** to 100" C), Dulpngand Petit, i. 01 80180 
Do. from 212° to 392° F. (100* to 200' C), do. 1.0184331 

Do. from 392" to 572° F. (200" to 300'* C), do. i. 0188679 

Do. from 32** to 212" F. ( o" to 100' C), Regnault, 1.0181530 



Expan- 
sion. 



V 



55-5 
V53 



Expansion of Gases by Heat. 

Gases are divisible into two classes — permanent gases and vapours. 
Gases for which great pressure and extremely low temperatures are neces- 
sary to reduce them to the liquid form, are called permanent gases, and 
those which exist in the fluid state under ordinary temperatures, are called 
vapours. 

The influence of heat in expanding a permanent gas maintained under a 
constant pressure, is such that, for equal increments of temperature, the 
increments of volume by expansion are also equal or very nearly equal ; in 
Other words, the gas expands uniformly, or very nearly uniformly, in pro- 
portion to the rise of temperature. 

Again, it has been observed that when the volume of permanent gases is 
maintained constant, the pressure increases uniformly, or nearly uniformly, 
with an increase of temperature. 

A perfect or ideal gas is one which, under a constant pressure, expands 
with perfect uniformity in proportion to the rise of temperature; and of which, 
also, when confined to a constant volume, the pressure increases with per- 
fect uniformity in proportion to the rise of temperature. 

When the temperature of atmospheric air is raised from 3 2** to 212° F., 
the following are the total increments of volume or of pressure, according 
to the treatment, as determined by Regnault, when the volume at 32** is 
taken as i : — 



EXPANSION OF GASES BY HEAT. 



343 



Air. Tbmpsraturb. Incrbmsnt. 

Pressure constant 32° to 212" F Volume increased from i to 1.3670. 

Volume constant 32° to 212° F Pressure increased from i to 1.3665. 

Showing that the increase of pressure, .3665, with a constant volume, is 
sensibly the same as, though less than, the expansion or increase of volume, 
.3670, when the pressure is constant 

The table No. 1 1 1 gives the expansion and the increase of pressure, for 
several gases, when raised from 32° to 212'' F.: — 

Table No. in. — Expansion and Pressure of Gases raised from 

32°t0 2I2'*F. 

(Renault.) 



Gasbs. 


Expannon of Gases under x Atmosphere. 


Increase of Pressure 

of Gases under a Con- 

tant Volume. 


Final Volimic at aia' F. 


Estpansion at sza* F., 
in Common Fractions. 


Fizud Pressure at 2x3*. 


Atmospheric Air 

Hvdroflfen 


Initial volume at 32*=!. 

1.3670 
1. 3661 

1.3669 
I.3710 

I.3719 
1-3877 
13903 


Initial volume at 39*= z. 

/a.73 
;A.73 

/2.71 

/a. 7a 
Va.6i 
Va.6o 


Initial pressure at 32*= X. 

1.3665 
1.3667 
1.3668 
1.3667 
1.3688 
1.3676 
1.3829 

1.3843 


Nitrocren 


Carbonic Oxide 

Carbonic Acid 

Nitrous Oxide 

Cvanocen 


Sulphurous Acid 



Table No. 112. — Expansion of Gases raised from 32° to 212° F., 
under Different Pressures, these Pressures remaining Con- 
stant FOR EACH Observation. 

(Regnault.) 



Gas. 


Pressure. 


Volume at 2x2*. 


Air 

• 


Millimetres. 
760 

2525 
2620 


Atmospheres. 
I. GO 
3.32 
3.45 


Volume at 32' F. = x. 
1.36706 
1.36944 
1.36964 


Hydrogen 


760 
2545 


1. 00 
3.35 


I.36613 
I.36616 


Carbonic Acid 


760 
2520 


1. 00 
332 


1.37099 
1.38455 


Sulphurous Acid 


760 
980 


1. 00 
1. 16 


1.3903 
1.3980 



344 HEAT. 

The first part of the table, No. iii, on the expansion of gases by heat, 
shows that the expansion, which is a little more than a third of the initial 
volume, is nearly die same for air, hydrogen, and carbonic oxide, which are 
sensibly perfect gases, and have never been liquefied. On the contrary, 
carbonic acid, cyanogen, and sulphurous acid have a greater enlargement of 
volume than those gases, and they are gases which may easily be liquefied. 

The second part of the table, column 4, shows that, when the volume 
is constant, the pressure is increased nearly in the same proportion as the 
volume is increased, when the pressure is constant. This nearness of the 
proportions is particularly close in the cases of the three sensibly perfect 
gases, — ^air, hydrogen, and carbonic oxide. 

The next table. No. 112, contains the results of Regnault's experiments 
on the expansion of gases from 32° to 212° F., under various constant 
pressures of from i to 3^ atmospheres. It is shown that the expansions of 
air and of hydrogen are sensibly the same, whether the constant pressure be 
I atmosphere or between 3 and 4 atmospheres ; whilst the expansions of 
carbonic acid and sulphurous acid are higher at the higher pressure. 

The deductions of Regnault, from his experiments, comprised the 
following principles : — 

That for air, and all other gases except hydrogen, the coeffifcient of 
dilatation, or the increment of expansion for one degree rise of temperature, 
increases to some extent with their density. 

That all gases possess the same coefficient of dilatation when in a state 
of extreme tenuity ; but that this law is departed from as gases become 
dense. 

Adopting, nevertheless, the mean of the results of the experiments of M. 
Regnault and of M. Rudberg, the expansion of one volume of air measured 
at 32** F., when heated to 212° F., under a constant pressure, will, for future 
calculation, be taken as equal to 0.365; the ratio of the initial to the 
expanded volume being as i to 1.365. As the expansion is uniform with 
the rise of the temperature through 180°, the expansion for each degree 
Fahr. is — 

.365-^180 = —^ - , 
493-2 

the volume at 32° F. being = i. The same uniform rate of expansion holds 
sensibly for temperatures higher than 212°; it has been verified experi- 
mentally up to 700° F., under one atmosphere. It is inferred that, con- 
versely, air would contract uniformly under uniform reductions of temperature 
below 32^* F., until, on arriving at 493°. 2 below the freezing point, or 
46 1**. 2 F. below zero, the air would be reduced to a state of collapse, 
without elasticity. This point in the Fahrenheit scale has thus been 
adopted as that of absolute zero, standing at the foot of the natural scale of 
temperature; and the temperature, measured from absolute zero, or 
— 461^2 F., is called the absolute temperature. 

Accordingly, if a given weight of air at 0° F. be raised in temperature to 
+ 461° F., under a constant pressure, it is expanded to twice its original 
volume; and if heated from 0° F. to twice 461°, or 922°, its original 
volume is trebled. 

In briefi ^^ follows that, sensibly. 



EXPANSION OF GASES BY HEAT. 345 

ist The pressure of air varies inversely as the volume when the tempera- 
ture is constant 

2d. The pressure varies directly as the absolute temperature when the 
volume is constant. 

3d. The volume varies as the absolute temperature when the pressure is 
constant. 

4th. The product of the pressure and volume is proportional to the 
absolute temperature. 

The absolute zero-point by different thermometrical scales is as follows : — 

Reaumur - 2 1 9°. 2 

Centigrade -274° 

Fahrenheit - 46i°.2 

To simplify calculation, the decimal is usually dropped from the Fahrenheit 
temperature, which is taken as -461°. 

The foregoing laws do not apply exactly to the expansion and contraction 
of the more easily condensable gases, for these, as they approach the point 
of liquefaction, become sensibly more compressible than air. Oxygen, 
nitrogen, hydrogen, nitric oxide, and carbonic oxide follow the same ratio of 
compression as that of air, being incondensable gases, at least as far as 100 
atmospheres of pressure. Sulphurous acid, ammoniacal gas, carbonic acid, 
and protoxide of nitrogen, which have been proved, on the contrary, to be 
condensable, become sensibly more compressible than air when they are 
reduced to one-third or one-fourth of their original volume at atmospheric 
pressure. Carbonic acid, under five atmospheres, occupies only 97 per 
cent of the volume which air occupies under the same pressure ; and under 
forty atmospheres, near the condensing point, it occupies only 74 per cent, 
or barely three-fourths of the volume of air at the same pressure. It has, 
nevertheless, been established that all gases, at some distance from the 
point of maximum density for the pressure, beyond which point they must 
condense, sensibly follow the first law above recited, according to which the 
pressure and the density vary directly as each other, when the temperature 
is constant With such limitations, they rank as perfect gases. 

The table No. 113 contains examples of the progressive pressures required 
to compress air, nitrogen, carbonic acid, and hydrogen, into one-twentieth 
of their original volumes, founded on experiments made by M. Regnault 
The pressures are expressed in metres of mercury, the pressure of a column 
of mercury one metre high being equal to 19.34 lbs. per square inch. The 
table shows that hydrogen is the most perfect type of gaseity. When 
compressed to a twentieth of its original volume, it supports something 
more than twenty times the original pressure. Air, on the contrary, requires 
a quarter of a metre less than 20 metres of pressure ; nitrogen requires a 
fifth of a metre less; and carbonic acid, like an overloaded spring, 3^ 
metres less. 



346 



HEAT. 



Table No. 113. — Compression of Gases by Pressure under a 

Constant Temperature. 



1 

Ratio of the 




Pressiuv in Metres of Mercury for 


original volume 
to the reduced 
















volume. 


Air. 


Nitrogen. 


Carbonic Acid. 


Hydrogen. 




Metres. 


Metres. 


Metres. 


Metres. 


I 


1.000 


1. 000 


1. 000 


1. 000 


2 


1.998 


1.997 


1.983 


2.001 


4 


3-987 


3.992 


3.897 


4.007 


6 


5-970 


5.980 


5-743 


6.018 


8 


7.946 


7.964 


7-519 


8.034 


10 


9.916 


9-944 


9.226 


10.056 


12 


11.882 


II.919 


10.863 


12.084 


14 


13-845 


13.891 


12.430 


I4.II9 


16 


15.804 


15.860 


13.926 


16.162 


18 


17-763 


17.825 


15.351 


18.2II 


20 


19.720 


19.789 


16.705 


20.269 



Note, — 20 metres of mercury are equal to a pressure of 386.8 lbs. per square inch, or 
26.3 atmospheres. 

Relations of the Pressure, Volume, and Temperature of Air 

and other Gases. 

In accordance with the relations of pressure, volume, and temperature 
above stated, it is found that air and other perfect gases, and, within 
practical limits, the permanent gases generally, are expanded by heat at the 
rate of V461 part of their volume at 0° F. for each degree of temperature, 
under a constant pressure. If the volume at the freezing point, 32° F., be 
taken as the point of departure, the denominator of the fraction is 
461° + 32° = 493°, and the expansion is at the rate of ^453 part of the 
volume at 32° F. for each degree of temperature. In general, for any- 
other initial temperature the denominator of the fraction showing the rate 
of expansion for each degree is found by adding 461° to the initial tempera- 
ture. But, for convenience of calculation, the initial temperature is usually- 
taken at 0° F. 

Similarly, the pressure of air having a given constant volume, is increased 
by heat at the rate of 7461 part of the pressure at 0° F. 

The fraction of expansion when the pressure is constant, and the fraction 
of pressure when the volume is constant, for each degree of temperature by 
Fahrenheit's scale above o^, is, then. 



461' 

and the same fraction expresses the rate of contraction of volume for each 
degree of temperature below o® F. 

A number of proportions and rules for the relations of the pressure, 
volume, and temperature of a constant weight of a gas are readily deduced 
from the above defined ratios. 



PRESSURE, ETC., OF AIR AND OTHER GASES. 347 

 

1. When the pressure is constant, the volume varies as the absolute 
temperature; or, 

V : V: : / + 461 : /' + 461, and 
v/_v ^ + 461 . / \ 

" 7T^' ^^ 

in which V is the volume of the air or other gas at the temperature /, and 
V is the volume at the temperature /^. Whence the rule — 

Rule i. To fi?id the volume of a constant weight of air or otJier permanent 
gas, at any other temperature, when the volum^ at a given temperature 
is known, the pressure being constant. Multiply the given volume by the 
new absolute temperature, and divide by the given absolute temperature. 
The quotient is the new volume. 

Note, — The absolute temperature is found by adding 461° to the 
temperature indicated by the Fahrenheit thermometer. 

A common case of the above rule is, air at the mean temperature, 62** F., 
and mean atmospheric pressure, 14.7 lbs. per square inch. The increased 
volume, by expansion by heat, taking the initial volume = i, is found by 
substitution and reduction to be as follows : — 

v'jy^^2 (2) 

523 

Rule 2. To find the increased volume of a constant weight of air, of 
which the initial volume =1, taken at 62° F., heated under the ordinary 
atmospheric pressure of 14.7 lbs. per square inch, to a given temperature. 
To the given temperature add 461, and divide the sum by 523. The 
quotient is the increased volume by expansion. 

2. When the temperature of a constant weight of air, or other gas, is 
constant, the volume varies inversely as the pressure ; or, 

V : V : : /' : /, and 
V' = V^; (3) 

in which V and V are the volumes respectively at the pressures/ and/'. 

Rule 3. To find the volume of a constant weight of air or other permanent 
gas, for any pressure, when the volume at a given pressure is known, 
the temperature remaining constant. Multiply the given volume by the 
given pressure, and divide by the new pressure. The quotient is the new 
volume. 

3. When the pressure and temperature of a constant weight of air or 
other gas both change, the volume varies in the compound ratio of the 
absolute temperature directly, and the pressure inversely ; or, 

V : V ::/(/ + 46i):/(/' + 46i); 

or V'/ (/ + 46i) = V/ (/'+461), and 

V/-.v /(^ + 46i) , V 



34^ HEAT. 

Rule 4. To find the volume of a coftstant weight of air or other permanait 
gas for any other pressure and temperature, when the volume is known 
at a given pressure and temperature. Multiply the given volume by the 
given pressure, and by the new absolute temperature, and divide by the 
new pressure, and by the given absolute temperature. The quotient is the 
new volume. 

4. When the volume and temperature of a constant weight of air or 
other gas both change, the pressure varies in the compound ratio of the 
absolute temperature directly, and the volume inversely. 

/ :/ : : V'(/ + 46i) : V(/' + 46i); 
or V'/(/ + 46i) = V/ (/' + 461), and 

^ ^V'(/+46i) ^^^ 

Rule 5. To find the pressure of a constant weight of air or other permanent 
gas for any other volume and temperature, when the pressure is known for 
a given volume and temperature. Multiply the given pressure by the 
given volume, and by the new absolute temperature, and divide by the 
new volume, and by the given absolute temperature. The quotient is the 
new pressure. 

For the common case, when the initial temperature is 62® F., and the 
initial pressure is 14.7 lbs. per square inch, the formula (5) becomes, by 
substitution and reduction, 

v=VJ^+_46xi (6) 

^ 35-58 V' ^ > 

Rule 6. To find the pressure of a constant weight of air or other gas taken 
at 62° F., and at 14.7 ibs, pressure per square inch, with a given volume, for 
any other volume and temperature. Multiply the initial volume by the 
final temperature plus 461, and divide the product by the final volume, 
and by 35.58. The quotient is the new pressure in lbs. per square inch. 

When the volume is constant, with an initial temperature of 62** F., and 
an initial pressure of 14.7 lbs. per square inch, the above formula (6) is 
simplified thus : — 

p'=(±^ ^ J 

^ 35-58 ^^' 

Rule 7. To find the pressure of a constant weight of air or other gas takefi 
at 62° F., and at 14.7 lbs. pressure per square inch, with a constant volume, 
for a given temperjiture. Add 461 to the given temperature, and divide 
the sum by 35.58. The quotient is the pressure in lbs. per square inch. 

5. The mutual relations of pressure, volume, and temperature are con- 
densed in the following formula : — 

V/*/ + 46i, {a) 

the product of the volume and pressure of a constant weight of air being 
proportional to the absolute temperature. And, as that product bears 
always the same ratio to the absolute temperature, an equation may be 



PRESSURE, ETC., OF AIR AND OTHER GASES. 



349 



formed between them by multiplying the absolute temperature by a 
coefficient, which may be put = a. Then — 

V/ = ^ (/+461); {b) 

that is, the product of the volume and pressure of a constant weight of air 
or other permanent gas, is equal to the absolute temperature multiplied by 
a constant coefficient, which is to be determined for each gas according to 
its density. 

Special Rules for One Found Weight of a Gas. 

The application of formula {b) to a particular constant weight of gas, will 
suffice for many purposes. Let the constant weight be one pound of gas. 
To settle the coefficients for the different gases, take, for example, the 
temperature 32° F., giving an absolute temperature of 493**, and the pressure 
one atmosphere, or 14.7 lbs. per square inch. The volume of one pound 
of air at this temperature and this pressure is as before stated, 12.387 cubic 
feet Substitute these values for V, /, /, in the formula (^), then — 

12.387 X 14.7 =ax 493, 
whence the coefficient, a, for air is — 

^ = -36935* or 



2.7074 



and the formula {b) becomes, for air, 

/ + 461. 



V/ = 



2.7074 



(O 



Table No. 114. — Of Coefficients or Constants, «, in the Equation 
{b) FOR THE Relations of the Volume, Pressure, and Tem- 
perature OF Gases; namely, ^ p - a (/ + 461). 



> 

Name of gas. 


Volume of one pound of 

gas, at 32** F., under 

one atmosphere. 


Value of coefficient a. 


Hydrogen 

Gaseous steam 


cubic feet. 
178.83 

19-913 

12.753 
12.580 

12.387 

11.205 

8.157 

8.IOI 

4.777 
1.776 


5-33200, or 7o.x875 

0.59372, or 7^.6842 
0.38027, or 72.6297 

0.37506* or 7a.6662 

0.36935* or 72.7074 

0.33406, or 7a.5535 
0.24322, or 74.„x4 
0.24155, or 74.,3p9 
0.14246, or 77.019s 
0.05296, or 7,8.878 


Nitrogen 

defiant gas 

Air 

Oxygen 

Carbonic acid (ideal)* * . 

Do. do. (actual) 

Ether vapour* 

Vapour of mercury* 



* The densities are computed by Rankine for the ideal condition of perfect gas. 



3SO HEAT. 

that is to say, the volume of one pound of air, multiplied by the pressure 
per square inch, is equal to the absolute temperature divided by the 
constant 2.7074. 

To adapt the formula (d) for other gases, the respective coeflScients, or 
constants, are found in the same manner, in terms of the volume of one 
pound of each gas, at 32° F., under one atmosphere of 14.7 lbs. per square 
inch. They are given in table No. 114. 

6. The volume of one pound of air at any pressure and temperature is 
deduced as follows : — 

V=^+46i ^gj 

2.7074/ ^ ' 

Rule 8. — To find the volume of ofie pound of air, of a given temperature 
and pressure. Divide the absolute temperature by the pressure in lbs. per 
square inch, and by 2.7074. The quotient is the volume in cubic feet. 

For the ordinary case When the pressure is constant at 14.7 lbs, per 
square inch, the formula (8) becomes, by substituting and reducing, 

^ = lPo- ^9) 

Rule 9. — To find the volume of one pound of air under 14.7 lbs, pressure 
per square inch, at a given temperature. Add 46 1 to the temperature, and 
divide the sum by 39.80. The quotient is the volume in cubic feet. 

7. The pressiure of one pound of air of any volume, and at any tempera- 
ture, is found as follows : — 

^ = ^7^4V <'°) 

Rule 10. — To find tJie pressure of one pound of air, of a given temperature 
and volume. Divide the absolute temperature by the volume and by 2.7074. 
The quotient is the pressure in lbs. per square inch. 

8. The temperature of one pound of air of any volume and pressure is 
found as follows : — 

/= 2.7074 V/- 461 (11) 

Rule i i. — To find the temperature of one pound of air, of a given relume 
and pressure. Multiply the volume by the pressure in pounds per square 
inch, and also by 2.7074; subtract 461 from the product. The remainder 
is the temperature. 

9. The density of air is inversely as the volume, and is expressed by an 
inversion of the formula (8), for the volume; thus, putting D for the density, 
or the weight in pounds of one cubic foot of air — 

^ = '-7°747:j^ (") 

Rule 12. — To find the density of air, at a given temperature and pressure. 
Multiply the pressure in pounds per square inch by 2.7074, and divide 
by the absolute temperature. The quotient is the density, or weight in 
pounds of one cubic foot. 



VOLUME, DENSITY, AND PRESSURE OF AIR. 



351 



Table No. 115. — Volume, Density^ and Pressure of Air at various 

Temperatures. 



1 

1 


Volame of one pound of air at 


Density, or weight 


Pressure of a given weight of air 
having a constant volume. 


Temperature. 


constant atmospheric pressure. 


of one cubic foot of 


14.7 lbs. per square inch. 
Daium— Volume at 62"' F. = i. 


air at atmospheric 


Datum — Atmospheric pressure at 

62- f: = X. 




pressure. 


Fahrenheit. 


cubic feet. 


comparative 
volume. 


pounds. 


pounds per 
square inch. 


comparative 
pressure. 


0^ 


.11-583 


.881 


.086331 


12.96 


.881 


32 


12.387 


•943 


.080728 


13.86 


-943 


40 


12.586 


-958 


.079439 


14.08 


-958 


50 


12.840 


-977 


.077884 


14.36 


-977 


62 


13-141 


1. 000 


.076097 


14.70 


1. 000 


70 


13-342 


1.015 


.074950 


14.92 


1.015 


80 


13.593 


1.034 


.073565 


15.21 


1.034 


90 


13-845 


1-054 


.072230 


15-49 


1.054 


ICO 


14.096 


1.073 


.070942 


15.77 


1-073 


120 


14-592 


I. Ill 


.068500 


16.33 


I. Ill 


140 


15.100 


1.149 


.066221 


16.89 


1.149 


160 


15-603 


1. 187 


.064088 


17.50 


1. 187 


180 


16.106 


1.226 


.062090 


18.02 


1.226 


200 


16.606 


1.264 


.060210 


18.58 


1.264 


210 


16.860 


1.283 


-059313 


18.86 


1.283 


212 


16.910 


1.287 


.059135 


18.92 


1.287 


220 


I7.III 


1.302 


.058442 


19.14 


1.302 


230 


17.362 


1.321 


-057596 


19.42 


1. 321 


240 


17.612 


I -340 


.056774 


19.70 


1.340 


250 


17.865 


1-359 


.055975 


19.98 


1-359 


260 


18.I16 


1-379 


.055200 


20.27 


1-379 


270 


18.367 


1.398 


-054444 


20.55 


1.398 


280 


18.621 


1.417 


-053710 


20.83 


1.417 


290 


18.870 


1.436 


.052994 


21. II 


1.436 


300 


I9.I2I 


1-455 


.052297 


21.39 


1-455 


320 


19.624 


1-493 


.050959 


21.95 


1-493 


340 


20.126 


1-532 


.049686 


22.51 


1-532 


360 


20.630 


1-570 


.048476 


23.08 


1.570 


380 


21. 131 


1.608 


.047323 


23-64 


1.608 


400 


21.634 


1.646 


.046223 


24.20 


1.646 


425 


22.262 


1.694 


.044920 


24.90 


1.694 


450 


22.890 


1.742 


.043686 


25.61 


1.742 


475 


23-518 


1.789 


.042520 


26.31 


1.789 


500 


24.146 


1.837 


.041414 


27.01 


1-837 


525 


24.775 


i.«85 


.040364 


27.71 


•1.885 


550 


25-403 


1-933 


.039365 


28.42 


1.933 


575 


26.031 


1.981 


.038415 


29.12 


1.981 


600 


26.659 


2.029 


.037510 


29.82 


2.029 



352 



HEAT. 



• 




Table No. 


115 {continued). 






Volume of one pound of air at 


Density, or weight 


Pressure of a given weight of air 
having a constant volume. 




constant atmospheric pressure. 


of one cubic foot of 


Temperature. 


14-7 lbs. per square inch. 
Datum — ^Volume at 62" F. = i. 


air at atmospheric 


Datum — Atmosc 
6a' F. 


thenc pressure at 




pressure. 


= X. 


Fahrenheit. 


cubic feet. 


comparative 
volume. 


pounds. 


pounds per 
square inch. 


comparative 
pressure. 


650 


27.915 


2.124 


.035822 


31.23 


2.124 


700 


29.172 


2.220 


.034280 


32.63 


2.220 


750 


30.428 


2.315 


.032865 


34.04 


2.315 


800 


31.685 


2.41 1 


.031561 


35.44 


2.41 1 


850 


32.941 


2.507 


.030358 


36.85 


2.507 


900 


34.197 


2.602 


.029242 


38.25 


2.602 


950 


35.453 


2.698 


.028206 


39.66 


2.698 


1000 


36.710 


2.793 


.027241 


41.06 


2.793 


1500 


49.274 


3.749 


.020295 


55.12 


3.749 


2000 


61.836 


4.705 


.016172 


69.17 


4.705 


2500 


74.400 


5.661 


.013441 


83.22 


5.661 


3000 


86.962 


6.618 


.011499 


97.28 


6.618 



Note to Rules 8, 9, 10, 11, 12. — ^The coefficients or constants for other 
gases, in the application of the preceding five formulas and rules, are given 
in table No. 114. 

The table No. 115 contains the volume, density, and pressure of air at 
various temperatures from 0° to 3000° F., starting from 62° F. and 14.7 lbs. 
per square inch respectively as unity for the proportional volumes and pres- 
sures. The second column of the table, containing the volumes of one pound 
of air at different temperatures, was calculated by means of the formula (9), 
page 350. The third column, of comparative volumes, the volume at 62° 
F. being = i, was calculated by means of formula (2), page 347. The fourth 
column, of density, contains the reciprocals of the volumes in column 2, 
but it is calculable independently by means of formula (12), page 350. The 
fifth column, of pressures, due to the temperatures, was calculated by means 
of formula (7), page 348. The sixth column contains the pressures expressed 
comparatively, the atmospheric pressure, 14.7 lbs. per square inch, being 
taken as i. 

SPECIFIC HEAT. 

The specific heat of a body signifies its capacity for heat, or the quantity 
of heat required to raise the temperature of the body one degree Fahrenheit, 
compared with that required to raise the temperature of a quantity of 
water of equal weight one degree. The British unit of heat is that which 
is required to raise the temperature of one pound of water one degree, from 
32° F. to 33** R; and the specific heat of any other body is expressed by 
the quantity of heat, in units, necessary to raise the temperature of one 
pound weight of such body one degree. 

The specific heat of water is represented by i, or unity, and there are 
very few bodies of which the specific heat equals or exceeds that of water. 
Specific heats are, therefore, almost universally expressible by firactions 
of a unit. 



SPECIFIC HEAT OF WATER. 353 

It is necessary to fix a standard of temperature, such as the freezing 
point, for the datum of specific heat, as the specific heat of water is not 
exactly the same at different parts of the scale of temperatures, but increases 
in an appreciable degree, as well as in an increasing ratio, as the tempera- 
ture rises. For temperatures not higher than 80° or 90° F., the quantity of 
heat required to raise the temperature of water one degree is sensibly 
constant ; at 86° F., it is not above one-fifth per cent, in excess of that at 
the freezing-point. At 212° F., it is about \y^ per cent, in excess of that 
at 32° F. Above 212° F., it increases more rapidly; at 302% it is 2^ per 
cent, more than at 32°, and at 402°, it is 4^^ per cent. more. 

The average specific heat of water between the freezing and the boiling 
points is 1.005, or one-half per cent, more than the specific heat at the 
freezing point. 

It follows from the increasing specific heat of water, as the temperature 
rises, that the consumption of heat in raising the temperature is slightly 
greater expressed in units than in degrees of temperature. To raise, for 
example, one pound of water from 0° to 100° C, or from 32** to 212** F., 
there are required 100.5 ^- units, or 180.9 ^' units, of heat. 

The specific heats of water in the solid, liquid, and gaseous state are 
grouped as follows : — 

Ice 0.504 

Water i . 000 

Gaseous Steam 0.622 

showing that in the solid state, as ice, the specific heat of water is only half 
that of liquid water ; and that, in the gaseous state, it is a little more than 
that of ice, or barely five-eighths of that of liquid water. 

The specific heat of all liquid and solid substances is variable, increasing 
sensibly as the temperature rises, and the specific heats of such bodies, as 
tabulated, are not to be taken as exact for all temperatures, but rather as 
approximate average values, sufficiently near for practical purposes. The 
specific heat of the same body is, however, nearly constant for temperatures 
under 212° F. 

The specific heats of such gases, on the contrary, as are perfectly gaseous, 
or nearly so, do not sensibly vary with density or with temperature. 

For the same body, the specific heat is greater in the liquid than in the 
solid state. For example : — 

Liquid. Solid. 

Water (specific heat) i .000 o. 504 

Bromine „ o.iir 0.084 

Mercury „ 0.0333 0.0319 

M. Regnault has verified, by numerous experiments, the conclusion 
arrived at by previous experimentalists, that, for metals, the specific heats 
are in the inverse ratio of their chemical equivalents. Consequently the 
products of the specific heats of metals, by their respective chemical 
equivalents, are a constant quantity. The same rule holds good for other 
groups of bodies of the same composition, and of similar chemical constitu- 
tion. The specific heat of alloys is sensibly equal to the mean of those of 
the alloyed metals. 

The following are the specific heats of water for various tempera- 

88 



-thermometer, calculated 



. (i) 



tures from o° to 230° C, or 32° to 446° F., by the a 
by means of Regiiault's fonnula : — 

c- I + 0.00004 /+ 0.0000009/-; 

in which ^ is the specific heat of water at any temperature /, the specific 
heat at the freezing point being - .1. 

Table No. 116, — Specific Heat of Water. 











01 Specific 1 




Units of H«t 




fie Hon Bl He 








romihefnieiLng , I 


e given thefr. 






poinl 10 I he giv 




prrature. anc 


KIHUIK. 


F^n=„b.i.. 


Cecil, unki. 


Fahr.uml^ F™ 


ngpoLnI=.. 




3!° 


0,000 


0.000 I 


0000 




^t 


10.002 


18.004 ' 


0005 1 


0002 


68 


aO.oio 


36.018 


OD12 I 


0005 


86 


30.026 


54.047 1 


0020 I 


0009 


104 


40,051 


72.090 1 


0030 I 


0013 


122 


50.087 


90.157 I 


0042 1 


0017 


140 


60.137 


108.247 I 


0056 


0023 


IS« 


70.210 


126.378 1 


0072 1 


0030 


176 


80.282 


144.508 1 


0089 1 


0035 


194 


90.381 


162.686 I 


0109 1 


0042 


ai2 


100.500 


180.900 I 


0130 1 


0050 


230 


110.64. 


199.152 I 


0153 I 


0058 


248 


120.806 


217.449 I 


0177 1 


0067 


266 


130.997 


235-791 1 


0204 I 


0076 


284 


141.21S 


254-187 I 


0232 1 


0087 


302 


151.462 


272.628 1 


0262 I 


0097 


320 


161.741 


291.132 1 


0294 I 


0109 


338 


172.052 


309.690 I 


0328 I 




356 


182.398 


328.320 I 


0364 1 


0133 


374 


192.779 


347.004 1 


0401 1 


0146 


39* 


203,200 


365.760 I 


0440 1 


0160 


410 


213.660 


384-588 


0481 I 


0174 


428 


224.162 


403.488 1 


0524 1 


0.89 


446 


234.708 


422,478 I 


0568 1 


0204 



The Specific Heat of Air and other Gases, 

:ific heat, or capacity for heat, of permanent gases is sensibly 
r all temperatures, and for all densities. That is to say, the 
 heat of each g;as is the same for each degree of temperature. 
. Regnault proved that the capacity for heat was uniform for 
:s varying from -3o°C.to + 225°C. (- 22° to 437° F.); thus 
: heat for equal weights of air, at constant pressure, were as 



SPECIFIC HEAT OF AIR, ETC. 355 

Air between -30"* and + 10° C Specific heat, 0.2377 

Do. 10° and+ioo°C Do. 0.2379 

Do 100° and + 225"* C Do. 0.2376 

Average 0.2377 

The temperature is then without any^ sensible influence on the specific 
heat of air; neither has the pressure, so far as it has been subjected to 
experiment — from one to ten atmospheres — any influence on the magni- 
tude of the specific heat 

The specific heat of gases is to be observed from two points of view: — 
I St, When the pressure remains the same, and the gas expands by heat. 
' 2d, When the volume remains the same, and the pressure increases with 
the temperature. There is a striking diflference in the specific heat, or 
capacity for heat, according as it is measured under an increasing volume, 
or an increasing pressure. When the temperature is raised one degree, 
under constant pressure, with increasing volume, the gas not only becomes 
hotter to the same extent as when the volume remains the same and the 
pressure alone is increased, but it also expands 7493^ P^"^ ^^ ^^ volume 
at 32° F., and thus absorbs an additional quantity of heat in proportion to 
the work done by expansion against the pressure. It follpws that the 
specific heat of a gas at constant pressure is greater than that of the same 
gas under a constant volume; and though the former alone has been made 
the subject of direct experiment, the latter being of a diflicult nature for 
experimenters, yet the latter, which is properly the specific heat, is easily 
deducible from the former on the principle of the mechanical theory of 
heat 

When the volume of a gas is enlarged by expansion against pressure, the 
work thus done in expanding the gas may be expressed in foot-pounds by 
multiplying the enlargement of volume in cubic feet by the resistance to 
expansion in pounds per square foot Having thus found the work done 
in foot-pounds, it may be divided by Joule's equivalent, 772, and the 
quotient will be the expression of that work in units of heat. It becomes 
latent, or insensible to the thermometer, and is called the latent heat of 
expansion. It constitutes an expenditure of heat in addition to the heat 
that is sensible to the thermometer, and that raises the temperature. The 
sum of these two quantities of heat is that which has been observed in the 
gross by experimentalists, and which gives the specific heat at constant 
pressure. 

It follows that, when the specific heat at constant pressure is known, the 
specific heat at constant volume may be arrived at by subtracting the pro- 
portion of heat devoted to the enlargement of the volume from the total 
heat absorbed at constant pressure. The remainder is the proportion of 
heat necessary and sufficient to elevate the temperature when the volume 
remains unaltered, from which the specific heat at constant volume is 
deduced by simple proportion; thus — 

As the total heat absorbed at constant pressure, 

Is to the proportion of heat absorbed at constant volume, 

So is the specific heat at constant pressure 

To the specific heat at constant volume. 

For example, the specific heat of air at constant pressure and with in- 



I 



356 HEAT. 

creasing volume has been observed to be .2377, that of water being i. Let 
one pound of air at atmospheric pressure, and at 32° F., having a volume 
equal to 12.387 cubic feet, be expanded by heat to twice its initial volume, 
the pressure remaining the same. The absolute temperature, which is 32° 
+ 461 = 493** R, will be doubled, and the indicated temperature will be 
32 + 493 = 525° F. Thus, 493 degp-ees of heat are appropriated, and if the 
capacity for heat of the air were the same as that of water, 493 units of heat 
would be expended in the process of doubling the volume. But, as the 
specific heat is only .2377, or less than a fourth of that of water, the expen- 
diture of heat is just 493 x. 23 77=117. 18 units, and this quantity comprises 
the fraction of heat consumed in displacing the atmosphere and overcoming 
its resistance through a space of 12.387 cubic feet additional to the original 
or initial volume of the same amount. Now, the work thus done is equal 
to— 

12.387 cubic feet x 21 16.4 lbs. pressure per sq. foot = 26,216 foot-pounds; 

and dividing this by 772, Joule's equivalent, the work of enlarging or doub- 
ling the volume is found to be equivalent to 33.96 units of heat Deduct- 
ing these 33.96 units fi:om the gross expenditure, which is 11 7. 18 units, 
the remainder, 83.22 imits, is the proportion of heat required to raise the 
temperature through 493 degrees, under an increasing pressure simply, 
without increasing the volume ; and this remainder is the datum from which 
the proper specific heat of air is to be deduced. 

The distribution of heat thus detailed may be concisely exhibited 
thus : — 

Units. 

To double the temperature without adding to the volume.... 83.22 
To double the volume, in addition 33'9^ 



To double the temperature and double the volume at con- 
stant pressure 117. 18 

Now, as before stated, the specific heat at constant volume bears the 
same ratio to that at constant pressure, as the respective quantities, or units 
of heat, absorbed, do to each other, or as 83.22 and 11 7. 18; and it is found 
by simple proportion to be .1688 ; thus — 

117. 18 : 83.22 : : .2377 : .1688. 

The proper specific heat of air is then .1688, in raising the temperature 
without enlarging the volume, and it bears to the so-called specific heat of 
air, at constant pressure and with expanding volume, the ratio of i to 
1.408. 

This ratio, i to 1.408, deduced by means of the mechanical theory of 
heat, is practically identical with the ratio experimentally arrived at by M. 
Masson from the fall of temperature of a quantity of compressed air, which 
was liberated and allowed to expand back until it regained its initial pres- 
sure. The ratio he deduced firom his inverse experiment was i to 1.41; 
which is the ratio of 



I to 



\/3. 



SPECIFIC HEAT OF AIR, ETC. 357 

It may be added, by way of explanation, and to enforce the distinction, 
that though the pressure of a gas under constant volume rises with the 
tem^rature, — a. phenomenon which is analogous, at first sight, to that of the 
volume of a gas at constant pressure increasing with the temperature, — yet 
there is no expenditure of work in simply raising the pressure in the former 
case, while the volume remains unaltered ; whereas, in the latter case, there 
is an expenditure in increasing the volume, as has already been shown. 

To generalize the foregoing process, by which the specific heat of air at 
constant volume has been deduced from the specific heat for constant 
pressure; and to show its applicability for finding the specific heat of all 
gases at constant volume : — 

Given / = the initial temperature of the gas, in degrees Fahrenheit. 
„ /' = the final temperature to which the gas is raised. 

V = the initial volume of the gas, under one atmosphere of pres- 
sure, in cubic feet. 

V = the final volume of the gas, heated under constant pressure. 
„ A = the specific heat of the gas under constant pressure. 

Put A' = the specific heat of the gas under constant volume. 

H = the total heat expended at constant pressure, in units of heat. 
H' = the total heat expended at constant volume. 
H^ = the fractional quantity of heat expended in increasing the 
volume, at constant pressure; or the latent heat of expansion. 

To find the value of A'; then by proportion. 

And >4' = -^j-. 
NowH' = H-H^ 
And — = — ^ — , and, by substitution, 

Xx xl 



>> 



» 






>^=<2^' (a) 



Again, H = (Z' - /) x ^, 
And H^ = {V-t/) X 14.7 X 144-^772 
= (V-z/)x 2.742; 

And-^ Jjr^) • 

Substituting this value in equation (a) above, 

^, >4 (>4 (/'-/)- 2. 742 (V-z>)) . 
* A (/' - /) ' 

^^^,J{t'-t)-2^i_y{y-v) (^) 

Whence the following rule : — 

Rule i. To find the specific heat of a gas at constant volume, when the 
specific heat at constant pressure is given together with the initial and final 
temperatures due to given initial and final volumes under an atmosphere of 



3S8 HEAT. 

pressure. Multiply the difference of the initial and final temperatures by 
the specific heat at constant pressure. Also, multiply the difference of the 
initial and final volumes by 2.742. Find the difference of these two pro- 
ducts, and divide it by the difference of the temperatures. The quotient is 

ictfic heat of the gas at constant volume. 

lying the rule to the example of one pound of air at atmospheric 

■e, and at 32° F., doubled in volume by heat; A = .237 7, /'-/=493'', 

-»= 12.387 cubic feet. Then 

493 
cific heat of air at constant volume, as abeady found, 
comparative volumes of other gases are given in table No. 69, 
[6, of the Weight and Specific Gravity of G^s and Vapours. 

The Specific Heat of Gases for Eajual Volumes. 

specific heats of equal volumes of gases are deducible from their 
: heats proper, — which are for equal weights. The greater the 
, the less is the volume, and the greater the weight of gas that is 
iry to equal in volume a lighter gas; it is greater, in fact, in propor- 

the density. 

ce the following rule: — 

E 2. To find the specific heat of a gas for equal volumes of the gas 

air. Multiply the specific heat of the gas, that is, the specific heat 

al weights of the gas and air, by the specific gravity of the gas. The 

t is the specific heat for equal volumes. 

, — The specific heat for equal volumes may be found for constant 

e, and for constant volume, in terms respectively of the specific heat 

il weights at constant pressure and constant volume. 

ILES OF THE SPECIFIC HeAT OF SOLIDS, LIQUIDS, AND GaSES. 

annexed table. No. 117, contains the specific heats of a number of 
classified for convenience of reference, into 

Metals, 

Stones, 

Precious Stones, 

Sundry Mineral Substances, 

Woods. 
ipears from the tables that the metals, generally speaking, have the 
lecific heat; ranging from bismuth, having a specific heat of .031, 
, which has a specific heat of from, .n to .13, and iridium, which 
greatest specific heat, namely, .189. 

;s show a specific heat of about .20, or a fifth of that of water. 
IS stones average less than that. 

the sundry mineral substances, glass, sulphur, and phosphorus 
^ about a fifth of the specific heat of water, and coal and coke 

ds average a half of the specific heat of water. 



SPECIFIC HEAT OF SOLIDS. 



359 



It is a useful practical conclusion, as Dr. Rankine remarks, that the 
average specific heat of the non-metallic materials and contents oif a furnace, 
whftier bricks, stones, or fuel, does not greatly diflfer from one-fifth of that 
of water. 

Of the liquids specified in the table No. ii8, it appears that all, with the 
exception of bromine, which has a specific heat of i.iii, have less specific 
heat than water. Olive oil has the lowest, — only .31 ; alcohol averages .65, 
and vinegar, .92. 

The table No. 1 1 9 of the specific heat of gases, contains, in the second 
column, their specific heat, for equal weights, at constant pressure, as 
determined by M. Regnault The third column contains the specific heat, 
for equal weights, at constant volume, calculated by means of the Rule i, 
above. The fourth and fifth columns contain the specific heat of gases, 
for equal volumes, at constant pressure, and at constant volume, arrived 
at by means of the Rule 2, above. 

There is a remarkable nearness to equality in the specific heat for equal 
volumes of air, oxygen, hydrogen, carbonic oxide, and nitrogen. It may 
be noted, in particular, that hydrogen, though it has fourteen times the 
specific heat of air for equal weights, and has barely a fourteenth of the 
density of air, has no more specific heat than air, for equal volumes. 



Table No. 117. — Specific Heat of Solids. 

(Authority, Regnault^ when not otherwise stated.) 






METALS, from 32° to 212° F. 

Bismuth , 

Lead 

Platinum, sheet 

Do. spongy 

Do. 32° F. to 212° F {Petit and Duiong) 

Do. 32° F. to 572° F. (300° C.) „ 

Do. at 2i2°F. ( ioo°C.) {PouilUt) 

Do. at 572°F. ( 3oo°C.) „ 

Do. at 932° F. ( 500° C.) „ 

Do. ati292°F. ( 700° C.) „ 

Do. at 1832° F. (1000° C.) „ 

Do. at 2192° F. (1200° C.) „ 

Gold 

Mercury, solid 

Do. liquid 

Do. 59° to 68° F. (15° to 20° C.) 

Do. 32° to 212° F. {Petit and Duiong) 

Do. 32° to 572° F. (300° C.) 

Tungsten 

Antimony 

Do. 32° to 212° F {Petit and Duiong) 

Do. 32° to 572° F. (300^0.) 

Tin, English 

Do. Indian 



Water at 32°=!. 



.03084 

•0314 

.03243 
.03293 

.0335 

.0355 

.0335 

.03434 

.03518 

.036 

.03718 

.03818 

.03244 

.0319 

.03332 

.029 

.033 

•035 
.03636 

.05077 

.0507 

.0547 

.05695 
.05623 



36o 



HEAT. 



Metals {continued). 

Cadmium 

Silver 

Do. 32** to 212° F {FeHt and Dulong) 

Do. 32*^ to 572° F. (300° C.) 

Pal ladi um 

Uranium 

Molybdenum 

Brass 

Cymbal metal 

Copper 

Do. 32° to 212° F {Petit and Duiang) 

po. 32° to 572° F. (300° C.) 

Zinc 

Do. 32° to 212° F {Petit and Dulong) 

Do. 32°t0 572°F.<3oo°C.) 

Cobalt 

Do. carburetted 

Nickel 

Do. carburetted 

Wrought iron 

Do. 32° to 212° F. {Petit and Dulong 

Do. 32? to 392° F. (200° C.) „ 
Do. 32° to 572° F. (300° C.) 
Do. 32° to 662° F. (350° C.) „ 
Steel, soft 

Do. tempered 

Do. Haussman 

Cast iron, white 

Manganese, highly carburetted 

Iridium 

STONES. 

Brickwork and masonry {Rankine) about 

Marble, gray 

Do. white 

Chalk, white 

Quicklime 

Dolomite (Magnesian limestone) 

PRECIOUS STONES. 

Sapphire 

Zircon , 

Diamond 

SUNDRY MINERAL SUBSTANCES. 

Tellurium 

Iodine 

Selenium 

Bromine 

Phosphorus, 50° to 86° F 



Water at 32"=!. 

1~ 



05669' 
05701 

0557 
061 1 

05927 

0619 

07218 

0939 ^ 
086 

09515 
094 

1013 

09555 
0927 

1015 

10696 

I1714 

10863 

III92 

II379 
1098 

115 
1218 

1255 
I165 

II75 
1 1848 

12983 

14411 

1887 

20 
20989 

21585 
21485 

2169 
21743 

.21732 

•14558 
.14687 

•05155 
.05412 

.0837 

.0840 

.1887 



SPECIFIC HEAT OF SOLIDS. 



361 



Sundry Mineral Substances {continued), 

Riosphorus, 32"* to 212° F 

Glass 

Do. flint 

Do. 32° to 212^ F {Petit and Dulong) 

Do. 32° to 572° F 

Sulphur 

Do. crystallized, natural 

Do. cast for two years 

Do. cast for two months 

Do. cast recently 

Chloride of lead 

Do. zinc 

Do. manganese 

Do. tin 

Do. calcium 

Do. potassium 

Do. magnesium 

Do. sodium 

Perchloride of tin 

Protochloride of mercury 

Nitrate of silver 

Do. barytes 

Do. potass 

Do. soda 

Sulphate of lead *. 

Do. barytes 

Do. potash 

Carbonaceous : — 

Coal 

Charcoal 

Coke of cannel coal 

Do. pit coal 

Coal and coke, average {Rankine) 

Anthracite, Welsh 

Do. American 

Graphite, natural 

Do. of blast furnaces 

Animal black 

Sulphate of lime 

Magnesia 

Soda 

Ice 

WOODS. 

Turpentine 

Pear tree 

Oak 

Fir 



Water at 32*= I. 



25034 
19768 

177 

19 
20259 

1776 

1764 

1803 

1844 

06641 

13618 

14255 

14759 
16420 

17295 
19460 

.214 to .230 

IO161 
06889 

14352 
15228 

23875 
27821 

08723 

1 1 285 

1901 

24111 

2415 
20307 

20085 

20 

20172 

201 

20187 

497 
26085 

19659 
22159 

23115 
504 



.467 
.500 

.570 
.650 



362 



HEAT. 



Table No. 118. — Specific Heat of Liquids. 



>i 



yy 



Mercury 

O live oil (Laplace and Lavoisier) 

Sulphuric acid, density 1.87. 
Do. do. 1.30. 

Benzine, 59° to 68° F 

Turpentine, 

Do. density .872 (Despretz) 

Ether, oxalic 

Do., sulphuric, density 0.76 {Dalton) 

Do. do. do. 0.715 {Despretz) 

Essence of juniper 

Do. lemon 

Do. orange 

Hydrochloric acid 

Wood spirit, 59° to 68° F 

Chloride of calcium, solution 

Acetic acid, concentrated 

Alcohol 

Do. density o. 793 (Da/ton) 

Do. do. 0.81 

Vinegar 

Water, at 32° F 

Do. at 212° F 

Do. from 32* 
Bromine 



» 



'°to 212° F. 



Water at 32"=!. 



.0333 
.3096 

.3346 
.6614 

•3932 
.4160 

.4720 

.4554 
.6600 

.5200 

.4770 

.4879 

.4886 

.6000 

.6009 

.6448 

.6581 

.6588 

.6220 

.7000 

.9200 

1. 0000 

1. 0130 

1.0050 

I. mo 



FUSIBILITY OR MELTING POINTS OF SOLIDS. 



363 



Table No. 119. — Specific Heat of Gases. 

Water at 32'' F. = i. 



Qas. 



Sulphurous acid 

Vapour of chloroform 

Carbonic acid 

Oxygen 

Air 

Nitrogen 

Carbonic oxide 

defiant gas 

Hydrogen 

Vapour of Benzine 

Acetic ether 

Vapour of alcohol 

Gaseous steam , 

Vapour of turpentine . . » 

Ammoniacal gas 

Light carburetted hydrogen 



Specific Heat for 
Equal Weights. 



At constant 
pressure. 



water = i. 

01553 
0.1568 

0.2164 

0.2182 

0.2377 

0.2440 

0.2479 

0.3694 

3.4046 

0-3754 
0.4008 

0.4513 
0.4750 
0.5061 

0.5080 
0.5929 



At constant 
volume. 

(Real speci- 
fic heat.) 



water = x. 

0.1246 
0.1438 

O.I 7 14 

0.1559 
0.1688 

o. 1 740 
0.1768 
0.2992 
2.4096 

0.3499 
0.3781 

0.4124 

0.3643 

0.4915. 
0.391 1 

0.4683 



Specific Heat for 
Equal Volumes. 



At constant 
pressure. 



air = ,2377, 
as in col. a 

0.3489 

0.8310 

0.3308 

0.2412 

0.2377 

0.2370 

0.2399 

0.3572 
0.2356 
I.OII4 
1. 2184 
O.717I 
0.2950 
2.3776 
0.2994 
0.3277 



At constant 
volume. 



air = .1688, 
as in col. 3. 

0.2799 

0.7621 

0.2620 

0.1723 

0.1688 

0.1690 

O.171I 

0.2893 

0.1667 

0.9427 

1. 1490 

0.6553 
0.2262 

2.3090 

0.2305 

0.2588 



FUSIBILITY OR MELTING POINTS OF SOLIDS. 

The metals are solid at ordinary temperatures, with the exception of 
mercury, which is liquid down to - 39° F. Hydrogen, it is believed, is a 
metal in a gaseous form. 

All the metals are liquid at temperatures more or less elevated, and they 
probably vaporize at very high temperatures. Their melting points range 
from 39 degrees below zero of Fahrenheit's scale, the melting, or rather the 
freezing, point of mercury, up to more than 3000 degrees, beyond the 
limits of measurement by any known pyrometer. Certain of the metals, 
as potassium, sodium, iron, platinum, become pasty and adhesive at 
temperatures much below their melting points. Potassium and sodium, 
which melt at temperatures between 130° and 200° F., can be moulded 
like wax at 62° F. Two pieces of iron raised to a welding heat, are 
softened, and readily unite under the hammer; and pieces of platinum 
unite at a white heat. 

The melting points of alloys do not follow the ratios of those of their 
constituent metals, so that it is impossible to infer their melting points from 
these data. A remarkable instance of the absence of this relation is afforded 
in the fusible metal consisting of five parts of lead, three of tin, and eight 
of bismuth, which melts at 212**^, the heat of boiling water, though the 



364 



HEAT. 



melting point, if it were an average of those of the component metals, 
would be about 520'' F. The addition of bismuth to mixtures of lead and 
tin has the effect of lowering the melting points. 

According to Professor Rankine, the melting point of ice is lowered by 
pressure, at the rate of 0.0000063° F. for each pound of pressure on the 
square foot. An atmosphere of pressure being 2 116 lbs. per square foot, 
the lowering of the melting point per atmosphere of pressure, is — 

0°. 0000063 X 2116 = 0° 0133 Fahrenheit. 

To lower the melting point one degree, a pressure of 75 atmospheres would 
be required. 

In the case of water, antimony, and cast iron, and probably other sub- 
stances, the bulk of the substance in the solid state exceeds that in the 
liquid state, as is evidenced by the floating of ice on water, and of solid 
iron on molten iron. The volume of water is to that of ice at 32° F., as 
I to 1.088; that is to say, that water, in freezing at 32° F., expands nearly 
9 per cent. 

The following table. No. 120, contains the melting points of metals, 
metallic alloys, and other substances : — 



Table No. 120. — Melting Points of Solids. 



VARIOUS SUBSTANCES {Pauillet, CluMdel, &c) 

Sulphurous acid 

Carbonic acid 

Bromine 

Turpentine 

Hyponitric acid 

Ice 

Nitro-glycerine 

Tallow..... 

Phosphorus 

Acetic acid 

Stearine : 

Spermaceti 

Margaric acid 

Wax, rough 

„ bleached 

Stearic acid 

Iodine 

Sulphur 



Melting Points. 



- 148° F. 
-108 

+ 9-5 

14 
16 

32 

45 
92 

112 

"3 
109 to 120 

120 

131 to 140 

142 

154 

158 
225 

239 



MELTING POINTS OF SOLIDS. 



365 



Table No, 120 (cofiiinued). 



METALS. 



Mercury 

Rubidium 

Potassium 

Sodium 

Lithium 

Tin 

Cadmium 

Bismuth 

ThaUium 

Lead 

Zinc 

Antimony 

Bronze 

Aluminium 

Calcium 

Silver 

Copper 

Gold, standard 

Gold 

Cast Iron, white 

i> » gray 

„ „ „ 2d melting.. 

„ „ with manganese.. 

Steel 

Wrought Iron, French 

Hammered Iron, English..,. 

Malleable Iron 

Cobalt 

Nickel 

Manganese 

Palladium 

Molybdenum 

Tungsten 

Chromium 

Platinum 

Rhodium •. 

Iridium 

Vanadium 

Ruthenium 

Osmium 



Melting Points. 



Pouillet, Claudel. 



Fahrenheit degrees. 

-39° 

+ 136 
194 

446 

504 

608 

680 

810 

1692 



(very pure) 1832 

2156 

(very pure) 2282 

1922 to 2012 

2012 

2192 

2282 

2372 to 2552 

2732 

2912 



Wilson. 



Fahrenheit degrees. 

101° 

144 

208 

356 

442 

442 
617 

773 
1150 

full red heat, 
full red heat 

1873 
1996 

2016 

2786 



Fusible in highest 
heat of forge. 

Not fusible in forge 
fire, but soften and 
agglomerate. 



Only fusible before 
the oxyhydrogen 
blow-pipe. 



366 



h£at. 



Table No. 120 {continued). 



ALLOYS OF LEAD, TIN, AND BISMUTH. 



Melting Points. 



No. 



I. 
2. 

3- 

4- 

5. 
6. 

7- 
8. 

9. 
10. 

II. 

12. 



Tin, 25 Lead. 
10 

5 

3 
2 



% 



2 

3 

4 

5 
6 









13. 4 Lead, 4 Tin, i Bismuth 

^4* 3 » 3 » ^ » 

15- 2 „ 2 „ I „ 

16. I „ I „ I „ 

17. 2 „ I „ 2 

18. 3 „ 5 „ 8 






HoltzapfTcL 

558° 
541 

482 
441 

370 

334 

340 

356 

365 

378 

381 
320 

310 

292 

254 
236 

202 



ClaudeL 



466' 

385 
367 
372 
381 



SUNDRY ALLOYS OF TIN, LEAD, AND BISMXTTH. 

3 Lead, 2 Tin, 5 Bismuth Ure 

^ » ' » 2 „ „ 

1 „ I „ 4 „ Claudd 

5 » 3 » 8 „ ; Ure 

2 „ 3 » 5 » Claudd 

^ » 4 » 5 » >> 

' » ' » » 

^ >> 3 » » 

2 )) I )> )) 

I , / Holtzapffel 

" ' " t C&w^a' 

3 >» ' » >» 

3 j» ' >> >» 



Melting Points. 



199 
201 
201 
212 
212 
246 
286 

334 

334 
360 

38s 
392 

552 



ALLOYS FOR FUSIBLE PLUGS. 



Softens at 



Melts at 



2 Tin, 2 Lead 
2 „ 6 „ 

2 » 7 » 
2 „ 8 „ 



365° F. 
372 

377^4 
395 >^ 



372^ F. 

383 
388 

406 to 410 



LATENT HEAT OF FUSION OF SOLID «ODIES. 



367 



Latent Heat of Fusion of Solid Bodies. 

WTien a solid body is exposed to heat, and ultimately passes into